\  txAAjfrvv-, 


ELEMENTS 


OP 


CRYSTALLOGRAPHY 


FOR  STUDENTS  OF  CHEMISTRY  PHYSICS 
AND  MINERALOGY 


BY. 


GEORGE  HUNTINGTON  WILLIAMS  PnD 

ASSOCIATE  PROFESSOR  IN  THE  JOHNS  HOPKINS  UNIVERSITY 


NEW  YORK 
HENRY    HOLT    AND    COMPANY 

1890 


:    COPYRIGHT,  1890, 

BY 

HENRY  HOLT  &  CO. 


ROBERT  DRUMMOKD, 

Printer, 
NEW  YORK. 


PREFACE. 


THE  present  book  is  the  outgrowth  of  a  long-felt 
personal  need  and  the  hope  that  a  concise  and  ele- 
mentary statement  of  the  general  principles  of  Crys- 
tallography may  prove  acceptable  to  other  students 
than  those  of  mineralogy. 

To  both  chemists  and  physicists  the  subject  is 
important,  although  the  information  in  regard  to  it 
embraced  in  text-books  and  lectures  on  chemistry  and 
physics  is  usually  inadequate.  Crystals  are  constantly 
employed  as  a  means  of  purifying,  recognizing  and 
tracing  the  relationships  between  chemical  compounds 
of  all  sorts ;  their  genesis  and  growth  offer  interesting 
problems  in  molecular  physics,  and  their  completed 
forms  furnish  material  for  the  study  of  elasticity,  co- 
hesion and  the  propagation  of  various  forms  of  radiant 
energy. 

To  the  geologist  in  every  field,  as  well  a  to  the 
mining  engineer,  Crystallography  is  the  starting-point 
to  a  knowledge  of  mineralogy,  while  to  the  student  of 
petrography  an  acquaintance  with  crystal  form  and  its 
relation  to  the  optical  properties  of  crystals  is  indis- 
pensable. 

Considerable  experience  has  convinced  the  writer 
that  Crystallography,  in  its  simplest  form,  is  well 

iii 


iv  PREFACE. 

calculated  to  arouse  the  student's  interest  for  its  own 
sake,  and  that  it  may  with  advantage  be  incorporated 
into  many  courses  where  detailed  instruction  in  miner- 
alogy is  impossible.  This  book  is  not  intended  as 
a  complete  treatise,  but  merely  to  furnish  so  much  in- 
formation on  the  subject  as  may  be  of  service  to 
students  of  other  but  allied  branches.  It  is,  however, 
hoped  that  it  may  also  be  of  use  as  a  skeleton  for  a 
more  exhaustive  presentation  of  the  entire  subject 
when  such  is  desirable. 

Such  a  purpose  as  this  is  sufficient  excuse  for  the 
omission  of  much  that  is  of  cardinal  importance  to 
Crystallography  as  a  whole.  Mathematical  treatment, 
the  formulae  necessary  for  the  calculation  of  constants 
and  symbols  from  measured  angles,  the  application 
of  spherical  projection,  and  all  descriptions  of  the 
construction  and  manipulation  of  crystallographic  in- 
struments must  be  sought  for  in  larger  works,  whose 
titles  are  given. 

Those  methods  of  presentation  which  experience  has 
shown  to  be  most  readily  grasped  by  the  beginner 
have  been  throughout  preferred  to  those  which  ad- 
vanced workers  may  consider  more  elegant  and  satis- 
factory. The  symbols  of  Weiss  are  taken  as  a  starting- 
point,  since  they  most  clearly  indicate  the  position  of 
a  plane  with  reference  to  the  crystallographic  axes. 
The  shortened  form  of  these  symbols,  suggested  by 
Naumann,  has  been  generally  employed,  although 
Miller's  index  symbols  are  written  beside  them,  in 
order  to  familiarize  the  student  simultaneously  with 
both  of  these  methods  of  notation.  Naumann's  sym- 
bols have  been  preferred  to  those  of  Dana,  because  of 
their  more  general  use  in  works  on  crystallography 


PREFACE.  V 

and  mineralogy ;  and  because,  to  any  one  familiar  with 
Naumann's,  Dana's  symbols  can  present  no  difficulty. 
In  subject-matter  this  little  book  makes  no  claim  to 
originality.  It  is  only  an  attempt  to  present  to  English 
students  a  clear  and  concise  statement  of  the  results 
secured  by  others.  In  plan  and  illustration,  the  ad- 
mirable treatise  of  Groth  (Physikalische  Krystallo- 
graphie,  Leipzig,  1885)  has  been  freely  used.  Sug- 
gestions and  figures  have  also  been  taken  from  other 
sources  whenever  possible.  My  grateful  acknowledg- 
ments are  especially  due  to  my  friend  Professor  S.  L. 
Penfield  of  New  Haven,  to  whose  generous  advice  and 
suggestion  any  value  which  this  book  may  possess  is 
in  no  small  degree  due. 

BALTIMORE,  July,  1890. 


ERRATA. 

p.  7,  5th  line  from  bottom,  after  p  insert  .  vii. 

8,  Fig.  6  AVrong.      Alternate  revolved  rows  should  be 
not 


25  (bis),  49  (bis),  50,  86  (bis),  105  and  109,  tor  secant  re&dsectant 

30,  3d  line  from  top,  for  l-oo  read  l-i 

a      oo  0     I    (Oil)  read 
oo  a  ooO     /    (110) 
oo  c  Poo    l-i  1 110 1  read 
c       Poo    l-i     011} 


32,  6th 


119,  7th  line  from  bottom,  for  •  A/A.7;  read  n  \hlld  \ 
128,  Fig.  202,  on  lower  riglit  plane,  for  n  read  R 
133,  last  line,  for  trapezoedrons  read  trapezoliedrons 

mPn   r 
138,  ±__. 


141,  8th  line  from  top,  for  sulpliarsenide  read  sulpharsenite 
141,  9th    "       "       "      "    sulphantimonide  read  sulphantimonite 
145,  Fig.  240,  riglit  lower  angle  of  spherical  triangle  should 

reach  only  to  the  dotted  line. 
153,  5th  line  from  bottom,  for  [(Fe,Mg)SiOJ  read 


157,  3d  line  from  bottom,  for  triphenylmethan  read 

tripJienylmetliane 

157,  3d  line  from  bottom,  for  (C6H5)CH  read  (C6H5)3CH 
166,  4th  "        "      top,  for  Poo  read  Pec 
169,  4th    "        "         "      "     campheroxim  read  camplioroxime 
176,13th"        "       bottom,  for  |100}  read  J010} 

185,  top  line,  for  what  read  that 

186,  for  G.  289  read  FIG.  289 
205,  Fig.  337,  straighten. 

208,  13th  line  from  bottom,  for  by  read  from 

223,10th."        "  "        first  figure,  for  1  read  0 

223,  12th     "       "  "        last       "_      for  1    "     1_ 

227,10th     "        "  "       for  K  \  3141  }  read  K\  1341} 

228,  for  Fig.  368  read  Fig.  363 

240,  2d  line  from  top,  for  Od'  =  OD'.  sin  a  read  Od'=  Od  .  sin  a 


BIBLIOGRAPHY. 


THE  following  list  of  references  may  be  found  of  use 
by  those  desiring  fuller  information  in  regard  to  the 
subjects  touched  upon  in  this  book  : 

I.  On  the  Molecular  Structure  of  Crystals. 

K.  J.  HATJY  :  Essai  d'une  theorie  de  la  structure  des  cristaux. 

Paris,  1784. 

M.  L.  FRANKENHEIM  :  System  der  Krystalle.    1842. 
CHR.  WIENER  :  Grundziige  der  Weltordnung.     1863. 
A.  BRAVAIS  :  Etudes  cristallographiques.    Paris,  1866. 
A.  KNOP  :  Molectilarconstitution  und  Wachsthum  der  Krystalle. 

Leipzig,  1867. 
L.  SOHNCKE  :  Entwickelung  einer  Theorie  der  Krystallstructur. 

Leipzig,  1879. 
P.  GROTH  :  Die  Molecularbeschaffenheit  der  Krystalle.    Munich, 

1888. 

O.  LEHMANN  :  Die  Molecularphysik.     2  vols.     Leipzig,  1888. 
A.  FOCK  :  Einleitung  in  die  chemische  Krystallographie.     Leipzig, 

1888. 
L.  WULFF  :  Ueber  die  regelmassigen  Punktsysteme — Zeitschrift 

fiir  Krystallographie  und  Mineralogie,  vol.  xni.  pp.  503- 

566.    1887. 
L.   SOHNCKE  :  Erweiterung  der  Theorie  der  Krystallstructur— 

ibid.  vol.  xv.  pp.  426-446.     1889. 

II.  On  Crystallography. 

K.  J.  HAUY  :  Trait  e  de  cristallographie.     2  vols.     1822. 
F.  E.  NEUMANN  :  Beitrage  zur  Krystallonomie.    1823. 
C.  F.  NATJMANN  :  Lehrbuch  der  reinen  und  angewandten  Krys- 
tallographie.    2  vols.     1830. 

vii 


Vlll  BIBLIOGRAPHY. 

W.  H.  MILLER  :  A  Treatise  on  Crystallography.     1839. 

C.  F.  NAUMANN:  Eleraente  der  theoretischen  Krystallographie. 

1856. 

V.  VON  LANG  :  Lehrbuch  der  Krystallographie.     1866. 
A.  SCHRAUF  :  Lehrbuch  der  physikalischen  Mineralogie.     Vol.  I. 

1866. 
F.  A.  QUENDSTEDT  :  Grundriss  der  bestimmenden  und  rechnenden 

Krystallographie.     1873. 

ROSE  AND  SADEBECK  :  Elemente  der  Krystallographie.     1873. 
C.  KLEIN  :  Einleitungin  die  Krystallberechnung.     1876. 
E.  MALLARD  :  Traite  de  cristallographie  geometrique  et  physique. 

Vol.  I.     1879. 

TH.  LIEBISCH  :  Geometrische  Krystallographie.     1881. 
H.  BAUERMAN  :  Systematic  Mineralogy.     1881. 

E.  S.  DANA  :  Text-book  of  Mineralogy,  2d  Ed.     1883. 

A.  BREZINA  :  Methodik  der  Krystallbestimmung.     1884. 
P.  GROTH  :  Physikalische  Krystallographie,  3d  Ed.    Leipzig,  1885. 
V.  GOLDSCHMIDT  :  Index  der  Krystallf  ormen  der  Mineralien.     Vol. 
I.     Berlin,  1886. 

F.  HENRICH  :  Lehrbuch  der  Krystallberechnung.     Stuttgart,  1886. 
V.  GOLDSCHMIDT  :  Ueber  Projection   und    graphisehe    Krystall- 
berechnung.    Berlin,  1887. 

M.  WEBSKY  :  Anwendung  der  Linearprojection  zum  Berechnen 
der  Krystalle.     Berlin,  1887. 

G.  WYROUBOFF  :  Manuel  pratique  de  cristallographie.    Paris,  1889. 

in.  On  Crystal  Aggregates  and  Irregularities. 

A.  WEISBACH  :  Ueber  die  Monstrositaten  tesseral  krystalisirender 

Mineralien.     1858. 
C.   KLEIN  :    Ueber    Zwillingsverbindungen   und   Verzerrungen. 

Heidelberg,  1869. 

A.  SADEBECK  :  Angewandte  Krystallographie.     Berlin,  1876. 
G.  TSCHERMAK:    Zur  Theorie  der  Zwillingskrystalle— Mineralo- 

gische    und    petrographische    Mittheilungen,   vol.    n.    p. 

499.     1879. 
E.  MALLARD  :  Sur  la  theorie  des  macles— Bulletin  de  la  societe 

mineralogique  de  France,  vol.  vm.  p.  452.     1885. 
H.  BAUMHAUER  :  Das  Reich  der  Krystalle.    Leipzig,  1889. 


CRYSTALLOGRAPHY. 


CHAPTEB  I. 
CRYSTAL  STRUCTURE. 

The  Crystal.  All  chemically  homogeneous  sub- 
stances, when  they  solidify  from  a  state  of  vapor,  fu- 
sion, or  solution,  tend  to  assume  certain  regular  poly- 
hedral forms.  This  tendency  is  much  stronger  in 
some  substances  than  in  others,  and  it  varies  widely 
in  the  same  substance  under  different  physical  condi- 
tions. 

The  regularly  bounded  forms  thus  assumed  by 
solidifying  substances  are  called  crystals.*  Their 
shapes  are  directly  dependent  on  the  nature  of  the 
substance  to  which  they  belong,  and  they  are  there- 
fore valuable  for  its  identification,  like  any  of  its  physi- 
cal properties. 

Crystal  forms  have  been  particularly  useful  as  a 
means  of  recognizing  and  classifying  the  mineral 

*  This  term,  which  we  now  apply  to  all  of  these  forms,  was  used 
by  the  ancients  exclusively  for  crystallized  silica  or  quartz,  in  allu- 
sion to  the  then  accepted  idea  that  this  substance  was  ice  rendered 
permanently  solid  by  the  action  of  intense  cold.  (Greek,  KpvcrraX- 
Ao£,  from  KpvoS,  frost;  Latin,  crystallus.}  This  theory  was  still  ac- 
cepted by  Paracelsus,  and  was  not  combated  until  the  beginning  of 
the  seventeenth  century. 


2  CRYSTALLOGRAPHY. 

substances  which  compose  the  earth's  crust;  and 
hence  an  accurate  knowledge  of  their  geometrical  and 
physical  properties  has  long  been  considered  as  in- 
dispensable to  the  mineralogist,  geologist,  and  mining 
engineer.  Now,  however,  such  a  knowledge  is  hardly 
less  useful  or  necessary  to  the  chemist  or  physicist, 
irrespective  of  any  interest  he  may  have  in  the  min- 
erals and  rocks. 

The  regular  external  form  of  a  crystal  is  its  most 
striking  feature,  and  the  only  one  that,  for  a  long 
time,  was  regarded  as  important  or  essential.  But 
we  now  know  that  this  form  is  only  an  outward  expres- 
sion of  a  regular  internal  structure.  A  study  of  the 
physical  properties  of  a  crystal  aids  us  very  much  in 
properly  understanding  the  meaning  of  its  external 
form. 

If  we  examine  ordinary  homogeneous  substances 
which  are  not  crystals  in  regard  to  their  physical  prop- 
erties, such  for  instance  as  their  elasticity,  hardness, 
cohesion,  light-transmission,  heat-conduction,  etc.,  we 
find  that  these  are  equal  in  all  directions.  Thus  a 
piece  of  glass,  when  struck,  will  break  with  equal 
readiness  along  all  surfaces,  and  it  will  exhibit  an 
equal  degree  of  hardness  wherever  it  may  be 
scratched. 

With  crystals,  however,  this  is  not  true.  In  them 
we  find  differences  of  elasticity,  hardness,  cohesion,  and 
other  physical  properties,  which  do  not  exist  in  homo- 
geneous substances  which  are  not  crystals.  As  the 
result  of  long  study  by  many  eminent  observers,  the 
fact  has  been  established  that  the  distribution  of  physi- 
cal properties,  like  those  above  enumerated,  is,  in 
crystals,  equal  along  aU  parallel  directions,  ivhile,  with  cer- 


CRYSTAL  STRUCTURE.  3 

tain  exceptions,  it  is  unequal  along  directions  which  are  not 
parallel.  This  important  fact  gives  us  the  clue  to  the 
essential  nature  of  the  crystal ;  for  it  implies  that 
both  the  regular  external  form  and  the  distribution  of 
physical  properties  are  alike  directly  the  outcome  of 
some  regular  internal  structure.  We  may  make  a  glass 
model  of  exactly  the  same  shape  as  a  crystal,  but  it  is 
not  a  crystal,  in  spite  of  its  form,  because  the  necessary 
internal  structure  is  absent. 

Crystallography.  In  its  broadest  sense  this  term  re- 
lates to  the  scientific  description  of  crystals  in  all  their 
aspects.  This  wider  usage,  however,  naturally  falls 
into  three  subdivisions — geometrical  or  morphological, 
physical,  and  chemical  crystallography.  Of  these  three 
subjects  only  the  first,-  for  which  the  general  term 
crystallography  is  still  in  a  stricter  sense  reserved,  is 
embraced  within  the  scope  of  this  book. 

The  regular  forms  exhibited  by  crystals  were  made 
a  subject  of  elaborate  study,  before  the  distribution 
of  their  physical  properties  received  attention.  In 
this  way  these  forms  were  found  to  obey  certain  laws 
which  rendered  their  mathematical  treatment  and 
classification  possible.  Hence  the  science  of  geomet- 
rical crystallography  developed  quite  independently. 

Now,  however,  that  physical  crystallography  has 
shown  the  complete  accord  between  the  forms  and 
physical  behavior  of  crystals,  as  well  as  the  direct  de- 
pendence of  both  on  a  regular  internal  structure,  we 
must,  if  we  would  fully  appreciate  the  real  significance 
of  crystal  form,  first  discover  what  we  can  of  the  na- 
ture and  mode  of  arrangement  of  the  crystal  particles, 
themselves  invisible. 

The  Elementary  Crystal  Particles.     The  modern  con- 


4  CRYSTALLOGRAPHY. 

ception  of  matter  is  that  it  is  composed  of  ultimate 
particles,  called  atoms,  which  are  always  in  a  state  of 
intense  vibration,  and  which  are  separated  from  each 
other  by  distances  vastly  greater  than  their  own  diam- 
eters. The  number  of  kinds  of  these  atoms  is  com- 
paratively small,  but  by  uniting  into  groups  of  varying 
size,  composition,  and  arrangement  (called  chemical 
molecules),  they  are  capable  of  producing  all  the  variety 
of  substances  which  compose  our  material  world. 

It  is  not  improbable  that,  as  the  ultimate  elements 
of  matter  (the  atoms)  unite  to  form  chemical  mole- 
cules, so  these,  in  their  turn,  may  unite  to  form  groups 
of  a  higher  order,  called  physical  molecules.  The 
former  may  be  thought  of  as  comparable  to  our  solar 
system,  composed  of  individual  planets  which  are 
united  to  a  single  group  by  the  attraction  of  gravi- 
tation, while  at  the  same  time  the  solar  system  as  a 
whole  is  but  a  single  unit  in  a  vastly  larger  group  of 
systems,  which  is  held  together  by  the  same  attractive 
force. 

The  chemical  molecule  is  the  unit  of  substance,  be 
cause  we  cannot  imagine  it  to  be  divided  without  alter- 
ing the  substance.  The  physical  behavior  of  crystals 
also  necessitates  units  of  structure,  or  elementary  par- 
ticles by  whose  regular  arrangement  the  crystal  is  built 
up.  As  such  units  we  may  assume  the  physical  mole- 
cules ;  and,  for  this  purpose,  it  is  immaterial  whether 
they  are  different  from  the  chemical  molecules  or  not. 

Whatever  the  true  size  and  nature  of  these  crystal 
units  (crystal  molecules,  physical  molecules  *)  is,  we 


*  The  word  molecule,  as  used  in  the  following  pages  of  this  book, 
should  be  understood  as  always  referring  to  the  physical  and  not  to 
the  chemical  molecules. 


CRYSTAL  STRUCTURE.  5 

can,  for  all  purposes  of  explaining  crystal  structure, 
regard  them  as  points  (their  centres  of  gravity)  sur- 
rounded by  ellipsoids  or  spheres  whose  size  and  form 
represent  the  sum  of  all  the  various  attractive  and  re- 
pellent forces  inherent  in  the  molecules.  All  the  crys- 
tal molecules  of  the  same  chemical  substance  under 
the  same  conditions  must  be  identical  in  size,  shape, 
and  in  the  distribution  of  forces  ;  for  different  sub- 
stances they  must  be  different,  while  for  the  same  sub- 
stance under  different  conditions  they  may  or  may 
not  be  different. 

Mode  of  Molecular  Arrangement  in  Crystals.  If  the 
crystal  elements  or  physical  molecules  of  a  given 
substance  possess  the  same  size  and  the  same  attrac- 
tive forces,  then,  in  case  these  molecules  are  perfectly 
free  to  act  and  react  upon  each  other,  they  must  all 
assume  a  similar  position  relative  to  one  another,  i.e., 
such  a  position  that  equivalent  directions  of  attraction 
and  repulsion  in  all  the  molecules  shall  be  parallel. 

To  illustrate  this,  let  us  assume,  as  the  simplest  pos- 
sible case,  that  the  distribution  of  attractive  forces  in 
the  physical  molecules  of  a  certain  substance  is  equal 
in  three  directions  at  right  angles.  Then  such  a  mole- 
cule may  be  graphically  represented  by  Fig.  1,  where 
the  three  equal  and  perpendicular  dotted 
lines  represent  the  intensity  and  direc-  \ 

tion  of  the  attractions  inherent  in  the  "      py 
molecule.     Now  if   a  great  number   of 
similarly  constituted   molecules  of  this  •*• 

kind   are  gradually  approaching,  while 
their  forces  are  entirely  free  to  react  upon  each  other, 
they  will  finally  arrange  themselves  in  parallel  posi- 
tions, and  at  equal  distances,  corresponding  to  the  cor- 


6 


CRYSTALLOGRAPHY. 


FIG.  2. 


ners  of  a  cube,  Fig.  2.  If  we  conceive  of  an  arrange- 
ment like  this  as  indefi- 
nitely extended,  we  see 
that  the  grouping  about 
any  molecule  must  be  the 
same  as  about  every  other; 
and  also  that  the  arrange- 
ment of  molecules  in  all 
parallel  planes,  and  along 
all  parallel  lines,  must  be 
the  same.  This  is  still 
more  distinctly  seen  in 

Fig.  3,  where  the  distribution  of  molecules  along  the 
line  el  c3  a3  is  evidently  different  from  that  along  c,  #3  as, 

or  a,  a2  a3 ;  so  also  the  ar- 
rangement of  molecules  in 
the  plane  e1  az  e»  is  different 
from  that  in  the  plane  al  a,  c9cw 
etc. ;  while  in  parallel  planes, 
like  al  az  ag  a, ,  b1  b3  b9  b, ,  etc., 
the  grouping  is  the  same. 

No  arrangement  of  mole- 
cules, in  which  they  are  not 
similarly  distributed  in  all 
parallel  directions,  or  where 
the  grouping  is  not  the  same 
about  each,  can  be  regarded 
as  possible  in  a  crystal. 
Thus  the  study  of  crystal 
structure  becomes  an  investigation  of  the  possible 
networks  of  points  in  space  which  satisfy  these  condi- 
tions. This  problem  has  been  dealt  with  by  various 
writers,  especially  by  Solmcke,  who  finds  that  all  such 


FIG.  3. 


CRYSTAL  STRUCTURE.  7 

arrangements  which  are  possible  (sixty-six  in  num- 
ber) fall  naturally  into  groups  whose  symmetry  cor- 
responds with  that  of  the  systems  to  which  all  crystal 
forms  belong.* 

Crystalline  and  Amorphous  Substances.  If  the  passage 
of  a  chemically  homogeneous  substance  from  the 
gaseous  or  liquid  into  the  solid  state  is  too  rapid  to 
allow  of  the  perfect  action  of  the  attractive  and  re- 
pellent forces  upon  each  other,  the  crystal  molecules 
may  become  fixed  while  their  parallel  orientation  is 
still  incomplete  or  even  while  it  is  wholly  wanting. 

Many  substances  show  by  their  physical  properties 
that  they  possess  no  regularity  of  molecular  structure 
whatever.  Such  substances  never  exhibit  character- 
istic polyhedral  forms,  and  are  therefore  said  to  be 
amorphous.  Substances  which  are  only  known  in  the 
amorphous  state  are  usually  of  indefinite  chemical 
composition,  like  coal,  amber,  or  opal.  Definite  chem- 
ical compounds  almost  always  possess  some  power  to 
crystallize,  though  certain  usually  crystallized  sub- 
stances may  be  made  to  assume  an  amorphous  form  by 
very  much  accelerating  their  rate  of  solidification,  e.g., 
many  silicates,  when  fused  and  rapidly  cooled,  form  a 
glass.  The  real  difference  between  amorphous  and 
crystalline  substances  is  therefore  internal  and  molec- 
ular. A  fragment  of  quartz  and  a  fragment  of  glass 
may  to  all  external  appearances  be  quite  alike,  but 

*  Those  desiring  further  information  on  this  subject  should  con- 
sult the  works  of  Frankenheim,  Bravais,  Sohncke,  Groth,  and  Wulff , 
cited  on  p v: ' 

Sohncke  has  recently  been  led  to  extend  his  original  theory  by  the 
assumption  that,  in  certain  cases,  there  may  be  in  a  single  crystal 
two  or  more  interpenetrating  networks  like  those  above  described. 
(Zeitschrift  filr  Krystallographie,  vol.  xiv.  p.  426;  1888.) 


8 


CR  YSTALLOGHAPIIY. 


they  still  possess  important  internal  differences.  In 
the  former  the  elasticity  is  equal  in  parallel  directions 
and  different  in  directions  not  parallel,  while  in  the 
latter  the  elasticity  is  equal  in  all  directions.*  In  the 
former  the  molecular  arrangement  is  regular ;  in  the 
latter  quite  irregular. 

If  we  symbolize  the  physical  molecule  by  a  sphere 
with  all  of  its  attractive  and  repellent  forces  resolved 
into  three  directions  not  at  right  angles  to  one  another, 
the  position  of  such  molecules  in  different  states  of 
matter  may  be  graphically  represented  by  the  follow- 
ing four  diagrams^  Fig.  4  shows  the  spheres  so  widely 
separated  as  to  be  wholly  without  each  other's  influ- 
ence and  therefore  free  to  move  in  any  direction,  as 


FIG.  4. 


FIG.  5. 


FIG.  6. 


FIG.  7. 


. 

in  a  gas  or  liquid.  Fig.  5  represents  a  solid  state, 
where  each  molecule  is  free  to  move  only  within  its 
own  sphere  of  attraction,  the  position  of  this  being 
conditioned  by  those  of  the  surrounding  molecules. 
In  this  case  the  orientation  of  the  molecules  JLS  com- 
plete, as  in  a  simple  crystal.  Fig.  6Nillustmtes  another 
solid  state  where  the  orientation  is  only  partial,  as  in 
the  case  of  twin  crystals.  Here  the  molecules  of  each 

*  This  property  of  amorphous  bodies  does  not,  of  course,  apply  to 
organized  substances,  like  wood,  which  are  not  strictly  homo- 
geneous, nor  to  others  whose  differences  in  elasticity  are  due  to  ex- 
ternal strain,  as  in  the  case  of  unannealed  glass. 


CRYSTAL  STRUCTURE.  9 

horizontal  row  have  two  of  their  axes  parallel  with 
those  above  and  below  them,  but  not  the  third.  Such 
a  position  would  be  reached  by  supposing  the  mole- 
cules of  alternate  rows  in  Fig.  5  to  have  been  revolved 
180°  about  a  normal  to  the  plane  of  the  paper.  Fig.  7 
represents  a  solid  state  where  there  is  no  regularity  of 
molecular  orientation,  as  in  the  case  of  amorphous 
substances. 

Strength  of  the  "  Crystallizing  Force."  The  molecular 
forces  which  tend  to  produce  a  regular  internal  struc- 
ture in  matter  as  it  slowly  solidifies  exert  themselves 
in  varying  degrees,  both  in  different  substances  under 
the  same  conditions  and  in  the  same  substance  under 
different  conditions.  Such  variations  may  be  called 
differences  in  the  strength  of  the  "  crystallizing  force." 

This  crystallizing  force,  while  it  probably  exists  to 
some  degree  in  all  substances  of  definite  chemical 
composition,  is  so  very  weak  in  certain  ones,  like  ser- 
pentine or  turquoise,  that  their  crystal  form  is  not 
definitely  known.  That  they  really  crystallize,  and 
are  not,  strictly  speaking,  amorphous  bodies,  is  shown 
by  their  optical  and  other  physical  properties,  although 
conditions  favorable  enough  for  the  production  of  their 
crystal  form  appear  never  to  be  fulfilled. 

Other  substances,  like  some  of  the  metallic  sul- 
phides, rarely  possess  a  well-defined  crystal  form. 
They  commonly  occur  in  what  is  termed  the  massive 
state,  i.e.,  in  crystalline  aggregates  which  show  little  or 
no  trace  of  crystal  planes. 

Still  other  substances,  like  calcium  carbonate  (cal- 
cite),  and  silica  (quartz),  possess  an  intensely  strong 
crystallizing  force,  and  are  rarely  found  except  in  well- 
defined  crystals. 


10  CRYSTALLOGRAPHY. 

Mode  of  Crystal  Growth.  Crystals  are  distinguished 
from  living  organisms  by  the  method  of  their  growth. 
While  the  latter  grow  from  within  outward  and  are 
conditioned  both  in  their  form,  size,  and  period  of  ex- 
istence by  the  internal  laws  of  their  being,  crystals 
enlarge  by  regular  accretions  from  without,  and  are 
limited  in  size  and  duration  only  by  external  circum- 
stances. 

Organisms  must  pass  through  a  fixed  cycle  of  con- 
stantly succeeding  changes.  Youth,  maturity,  and 
old  age  are  unlike  and  must  come  to  all  in  the  same 
order.  There  is,  furthermore,  in  nearly  all  living 
things  a  differentiation  of  organs,  limitation  in  the  ex- 
tent of  growth,  and  the  power  of  reproduction. 

In  crystals,  on  the  other  hand,  every  part  is  exactly 
like  every  other  part.  Our  very  definition  of  crystal 
structure  is  an  arrangement  of  particles,  the  same 
about  one  point  as  about  every  other  point ;  hence,  in 
one  sense,  the  smallest  fragment  of  a  crystal  is  com- 
plete in  itself. 

Moreover,  since  crystals  grow  by  the  addition  of 
regular  layers  of  molecules,  arranged  just  like  all 
other  layers,  we  can  set  no  limit  to  the  size  of  a 
crystal,  so  long  as  the  supply  of  material  and  condi- 
tions favorable  to  its  formation  remain  constant. 
There  is  in  fact  the  widest  divergence  in  the  size  of 
crystal  individuals  of  the  same  composition  and  struc- 
ture. Those  of  ultra-microscopic  dimensions  and  those 
many  feet  in  length  may  be  identical  in  everything 
but  size.  Both  are  equally  complete,  and  one  is  in 
no  sense  the  embryo  of  the  other.  As  a  rule,  the  size 
of  a  crystal  is  inversely  proportional  to  its  purity 


CRYSTAL  STRUCTURE. 


11 


and  perfection  of  form,  but  this,  as  will  be  seen  at 
once,  is  dependent  on  external  conditions. 

Finally,  the  individual  crystal,  unlike  the  individual 
organism,  will  remain  unchanged  so  long  as  its  sur- 
roundings are  favorable  to  its  existence. 

Crystal  Habit.  Since  the  growth  of  a  crystal  is  pro- 
duced by  the  addition  of  regular  layers  of  molecules, 
fthen,  if  at  any  time  this  growth  be  interrupted,  the 
'crystal  will  be  bounded  by  plane  surfaces  which  rep- 
resent the  position  of  such  molecular  layers.  The  par- 
ticular planes  possible  in  any  given  case  must  there- 
fore depend  upon  the  mode  of  molecular  arrangement, 
and  hence  upon  the  chemical  composition  of  the  crys- 


FIG.  8. 


FIG.  9. 


FIG.  10. 


tal.  Nevertheless  the  number  of  planes  possible  on  a 
crystal  is  very  much  greater  than  that  which  actually 
occurs  in  any  single  instance.  It  is  reasonable  to 
suppose  that  only  planes  passed  through  a  regular  net- 
work of  molecules  so  as  to  intersect  the  same  number 
at  equal  distances  along  all  parallel  lines  are  possi- 
ble crystallographic  planes ;  while  those  planes  will  be 
of  the  most  frequent  occurrence  which  intersect,  in 


12  CRYSTALLOGRAPHY. 

this  way,  the  greatest  number  of  molecules.  Thus  we 
may  have  crystals  identical  in  composition  and  in  all 
their  physical  properties,  but  bounded  by  very  differ- 
ent sets  of  planes,  all  of  which  are  equally  possible 
with  the  same  internal  structure.  Such  differences  in 
form  among  crystals  of  the  same  substance  condition 
what  is  known  as  crystal  habit. 

The  three  preceding  figures,  8,  9,  and  10,  represent 
crystals  of  calcium  carbonate  (calcite).  The  forms, 
though  apparently  so  unlike,  can  all  be  shown  to  be 
derived  from  the  same  molecular  arrangement.  Each 
presents  a  combination  of  certain  out  of  a  great  num- 
ber of  possible  planes,  and  therefore  exhibits  a  par- 
ticular habit  of  a  single  substance.* 

*  A  graphic  illustration  of  the  molecular  structure,  as  well  as  of 
the  habit  of  crystals,  may  be  advantageously  employed,  which,  in 
principle,  is  not  unlike  the  well-known  figures  of  Haiiy.  If  we 
represent  the  physical  molecules  by  any  small  spherical  bodies,  of 
nearly  the  same  size,  like  shot,  we  can  readily  see  how  it  is  possible, 
by  the  same  arrangement  of  these,  to  build  up  different  forms.  A 
square  of  such  bodies,  arranged  in  parallel  rows,  may  be  taken  as  a 
starting  point,  and  then  by  piling  others  upon  them,  as  cannon-balls 
are  piled,  a  symmetrical  four-sided  pyramid  is  produced.  If  the 
shot  be  made  to  cohere  by  dipping  them  in  shellac,  a  similar  pyramid 
may  be  built  up  on  the  other  side  of  the  base,  thus  forming  the 
regular  octahedron.  Again,  if  successive  horizontal  and  vertical 
layers  be  taken  away  equally  from  each  of  the  six  solid  angles  of 
the  octahedron,  this  form  is  seen  to  develop  gradually  into  the  cube, 
while  the  interior  structure  remains  unchanged.  Finally,  we  may 
use  each  of  the  six  faces  of  the  cube  as  a  base  for  the  erection  of  a 
quadratic  pyramid,  and  thus  the  dodecahedron  is  formed,  with  a 
structure  like  that  which  produced  the  other  two  figures. 

Such  models  as  these  admirably  illustrate  how  differences  of  habit 
may  result  from  the  same  molecular  arrangement,  as  well  as  how 
the  planes  of  one  form  may  replace. the  edges  or  angles  of  another. 
We  have  only  to  conceive  of  the  shot  as  too  small  to  be  visible,  and 
the  surface  produced  by  any  layer  becomes  a  crystal  plane. 


CRYSTAL  STRUCTURE. 


13 


Exactly  what  it  is  that  determines  the  habit  of  a 
crystal  is  not  known.  Crystals  formed  at  the  same 
time  generally  exhibit  the  same  habit,  but  this  is  not 
always  the  case.  Doubtless  many  slight  alterations 
in  external  conditions  at  the  time  of  formation  may 
be  influential  in  this  regard.  Certain  crystals  possess 
different  habits  at  different  periods  of  their  growth. 
In  the  case  of  transparent  substances  a  core  or  kernel, 
bounded  by  different  planes  from  those  on  the  exte- 
rior of  the  crystal,  is  sometimes  visible  in  the  interior 
(German,  Kernkry  stall).  Fig. 
11  represents  a  calcite  crystal 
of  rhombohedral  habit,  from 
Mineral  Point,  Wisconsin,  in 
the  centre  of  which  is  a 
darker  scalenohedron.  Flu- 
orspar also  shows  this  phe- 
nomenon. Experiments  with 
many  artificial  salts  appear 
to  indicate  that  the  presence  of  impurities  in  the  con- 
centrated solution  may  be  a  most  important  factor  in 
conditioning  the  habit  of  crystals.  For  example,  so- 
dium chloride  may  be  obtained  in  octahedrons,  instead 
of  the  usual  cubes,  when  crystallized  from  a  solution 
containing  sodium  hydroxide.  On  the  other  hand, 
alum,  which  usually  crystallizes  in  octahedrons,  can 
be  produced  in  the  form  of  cubes  from  alkaline  solu- 
tions. The  habit  of  crystals  of  Epsom  salts  (magne- 
sium sulphate)  is  also  modified  by  the  presence  of 
borax  in  the  solution.  (See  O.  Lehmann,  Molecular- 
physik,  vol.  I.  p.  300.) 

Another  most  important  variation  in  crystal  habit 
is  often  produced  by  what  is  known  as  distortion  of 


FIG.  11. 


14 


CB  TSTALLOGRAPHT. 


the  form.  While  crystals  are  increasing  in  size  by 
the  addition  of  layers  of  molecules,  it  will  rarely 
happen  that  the  concentration  of  the  mother-liquor 
will  be  so  evenly  balanced  on  all  sides  as  to  make  the 
growth  equally  rapid  in  all  directions.  Where  the 
most  material  is  supplied,  there  the  growth  will  be 
most  accelerated ;  while  the  size  of  a  given  plane  will 
be  relatively  diminished  in  proportion  as  it  grows 
from  the  centre  of  the  crystal.  It  thus  happens  that 
crystallographically  equivalent  planes  vary  much  in 
size  on  the  same  individual,  and  that,  in  this  way,  the 
symmetry  of  a  form  is  often  completely  disguised ; 
to  restore  it,  all  similar  planes  must  be  imagined 
as  at  the  same  distance  from  the  centre.  In  this 
way  an  ideal  form  is  derived  from  the  distorted  form. 
This  is  a  matter  of  so  much  importance  to  the  beginner 
that  it  may  be  made  the  subject  of  illustration. 


Fio.  12. 


FIG.  13. 


Fio.  14. 


The  cubic  crystal,  Fig.  12,  may  grow  most  rapidly 
in  one  direction,  becoming  prismatic,  Fig.  13 ;  or  in 
two  directions,  becoming  tabular,  Fig.  14,  without  los- 
ing its  character  as  a  cube  so  long  as  its  angles  remain 
90°. 

The  symmetrical  octahedron,  Fig.  15,  may  become 
distorted,  as  in  Fig.  16,  or  even  flattened  into  trian- 


CRYSTAL  STRUCTURE. 


15 


gular  plates,  Fig.  17,  as  is  frequently  the  case  with 
alum  crystals. 


Fio.  15. 


FIG.  16. 


Fio.  17. 


The  next  three  figures,  18, 19,  and  20,  show  the  same 
combination  of  planes  unequally  developed  on  three 
crystals  of  quartz. 


FIG.  18. 


FIG.  19. 


FIG.  20. 


The  occurrence  of  ideal  forms  in  nature  is  rather 
the  exception  than  the  rule ;  hence  the  constructing 
in  imagination  of  the  symmetrical  equivalents  of  more 
or  less  distorted  crystals  becomes  an  important  mat- 
ter for  practice.  For  this  purpose  it  is  quite  neces- 
sary that  the  student  first  familiarize  himself  with 
the  ideal  forms ;  and  crystal  models,  with  which  the 
study  of  crystallography  must  be  commenced,  are  on 


16  CRYSTALLOGRAPHY. 

this  account  generally  represented  as  symmetrical 
bodies.* 

Crystal  Individuals  and  Crystalline  Aggregates.  That 
portion  of  a  homogeneous  crystallized  substance  whose 
molecular  arrangement  is  throughout  the  same  along 
all  parallel  lines,  and  which  is  bounded  by  its  own 
characteristic  plane  surface,  is  called  a  crystal  indi- 
vidual. Such  an  individual  is  not  of  necessity  com- 
pletely bounded  by  crystal  planes,  since  there  is  gen- 
erally a  larger  or  smaller  point  of  attachment  to  other 
crystals.  There  must,  however,  be  enough  planes  to 
allow  of  the  restoration  of  the  complete  form.  Any- 
thing less  than  this  is  a  crystal  fragment  or  grain. 

The  union  of  two  or  more  crystal  individuals  pro- 
duces a  crystal  aggregate ;  while  a  mass  of  crystal 
grains,  devoid  of  their  characteristic  forms  and  closely 
packed  together,  may  be  termed  a  crystalline  aggregate. 

*  Models,  which  are  of  such  prime  importance  in  the  study  of  crys- 
tallography, are  most  cheaply  and  elegantly  made  in  Germany.  The 
principal  forms  in  all  systems  are  constructed  of  glass  by  F.  Thomas 
at  Siegen  in  Westphalia.  These  show  the  position  of  the  axes  inside 
by  colored  threads,  and  are  particularly  valuable  for  demonstrating 
the  derivation  of  hemihedral  and  tetartohedral  forms  as  well  as  differ- 
ent methods  of  twinning.  They  are  large  enough  to  be  well  suited 
for  class-instruction  ;  and,  considering  the  perfection  with  which  they 
are  made,  are  furnished  at  a  very  reasonable  price. 

Models  of  convenient  size  are  accurately  made  of  hard  wood  by 
various  German  firms.  The  best  may  be  had  of  Krantz  in  Bonn,  who 
will  furnish  collections  of  any  required  size.  W.  Apel  of  Gottingen 
also  manufactures  a  small  but  useful  set.  Catalogues  of  all  these 
firms  may  be  had  on  application.  (See  also  the  price-list  of  crystallo- 
graphic  apparatus  at  the  end  of  Groth's  Physikalische  Krystallogra- 
phie;  3d  ed.,  1885.) 

Cardboard  models  may  be  made  by  students  by  cutting  out  and 
bending  into  shape  the  outlines  furnished  by  various  authors.  (See 
Kopp,  Einleitung  in  die  Kry  stall  ographie;  1862.  Atlas.) 


CRYSTAL  STRUCTURE.  17 

This  distinction  may  be  made  clearer  by  the  two  fol- 
lowing figures,  21  and  22. 


FIG.  21.— CRYSTAL  AGGREGATE. 
(Quartz  from  Dauphin6,  France.) 


FIG.  2-2.— CRYSTALLINE  AGGREGATE. 
(Gabbro  from  Prato  near  Florence,  Italy.) 

Aggregates  may  further  be  homogeneous  and  hetero- 
geneous, according  as  they  are  composed  of  one  sub- 


18  CRYSTALLOGRAPHY. 

stance  with  but  one  kind  of  molecular  structure,  like 
marble,  or  of  two  or  more  substances  with  different 
internal  structures,  like  granite.  The  subject  of  crys- 
tal aggregates  is  more  fully  treated  of  in  Chap.  IX. 

Limiting  Elements  of  Crystals.  With  certain  excep- 
tions to  be  explained  beyond,  all  crystals  have  their 
limiting  planes  so  arranged  in  pairs,  that  to  every  one 
there  is  a  parallel  plane  on  the  opposite  side  of  the 
crystal. 

The  planes,  or  crystal  faces,  intersect  in  edges  and 
angles.  A  crystal  edge  is  the  line  of  intersection  of 
two  crystal  planes  ;  the  angle  which  such  an  edge  en- 
closes is  called  an  interfaciol  angle.  By  the  term  crystal 
angle  is  meant  the  solid  angle  in  which  three  or  more 
crystal  faces  meet. 

Similar  edges  are  those  in  which  similar  planes  in- 
tersect at  equal  angles  ;  similar  angles  are  those  en- 
closed by  the  same  number  of  planes,  similarly  ar- 
ranged and  meeting  at  the  same  inclination. 

It  is  convenient  to  remember  that  on  all  polyhe- 
drons, and  hence  on  all  crystals,  the  number  of  faces 
plus  the  number  of  solid  angles  is  equal  to  the  num- 
ber of  edges  plus  two. 


General  Principles  of  Crystallography.  Before  pro- 
ceeding to  the  description  of  the  different  groups  or 
systems  into  which  all  crystal  forms  are  classified,  it 
will  be  necessary  to  consider  certain  common  proper- 
ties which  such  forms  possess.  These  are  : 

1.  Constancy  of  corresponding  interfacial  angles  on 
all  crystals  of  the  same  substance. 

2.  Simple  mathematical  ratio  existing  between  the 


CRYSTAL  STRUCTURE.  19 

co-ordinates  of  all  planes  which  are  possible  on  crys- 
tals of  the  same  substance. 

3.  Symmetry. 

We  may  regard  the  expression  of  these  common 
characters  of  crystals  as  the  fundamental  laws  of 
crystallography,  and  their  explanation  will  form  the 
subject  of  the  following  chapter. 


CHAPTEE  II. 
GENERAL  PRINCIPLES  OF   CRYSTALLOGRAPHY. 

1.  LAW  OF  THE  CONSTANCY  OF  INTEBFACIAL  ANGLES. 

Statement  of  the  Law.  Hoivever  much  the,  crystals  of  the 
same  substance  may  vary  in  habit  and  in  the  relative  size 
and  development  of  similar  planes,  their  corresponding  in- 
terfacial  angles  remain  constant  in  value  ;  provided  that, 
first,  they  possess  identically  the  same  chemical  composi- 
tion, and,  second,  that  they  are  compared  at  the  same 
temperature. 

This  identity  of  corresponding  angles  on  crystals  of 
the  same  substance  is  clearly  a  necessary  consequence 
of  their  possessing  the  same  molecular  structure 
under  the  same  physical  conditions.  It  was,  however, 
observed  to  be  true  long  before  crystals  were  thought 
to  have  any  peculiar  structure,  and  was  first  formu- 
lated by  Steno  in  1669.  It  was  further  substantiated 
by  Rome  de  1'Isle  with  a  contact-goniometer  in  1783, 
and  finally  established  within  a  very  small  limit  of 
error,  after  the  invention  of  the  reflecting  goniometer 
in  1809. 

The  importance  of  this  law  consists  in  showing  that 
the  exact  values  of  crystal  angles,  even  more  than  the 
particular  shapes  of  the  crystals  themselves,  are  char- 
acteristic of  the  substance  composing  them,  for  these 
angles  remain  constant  in  spite  of  all  the  distortions 
to  which  the  forms  are  subject  (p.  14). 

20 


GENERAL  PRINCIPLES.  21 

Indeed,  they  may  serve  to  distinguish  substances 
which  are  in  all  other  outward  respects  quite  identical. 
For  instance,  the  angle  105°  5'  is  always  included  be- 
tween the  cleavage  faces  of  a  crystal  of  calcium  car- 
bonate, while  the  corresponding  angle  on  crystals  of 
magnesium  carbonate  is  107°  28'. 

The  essential  nature  of  the  interfacial  angles  renders 
more  intelligible  the  distortion  of  crystal  forms  de- 
scribed in  the  preceding  chapter,  since,  in  spite  of 
a  parallel  shifting  of  the  planes,  the  angles  between 
them  remain  unaltered. 

Measurement  of  Interfacial  Angles.  In  order  to  ascer- 
tain how  far  the  law  of  the  constancy  of  interfacial 


FIG.  23. 


angles  holds  good  it  is  necessary  to  accurately  meas- 
ure these  angles,  A  knowledge  of  their  exact  values 
is  furthermore  requisite  for  any  mathematical  treat- 
ment of  crystal  forms.  An  instrument  for  measuring 
crystal  angles  is  therefore  of  prime  importance  in  the 
study  of  crystallography,  and  is  called  a  goniometer. 


22  CRYSTALLOGRAPHY. 

The  simplest  form  of  such  an  instrument,  called  a 
contact-  or  hand-goniometer,  was  first  constructed  near 
the  end  of  the  last  century.  It  consists  of  two  arms 
(Fig.  23),  one  of  which  revolves  about  a  pivot  fastened 
to  the  other,  which  may  be  set  by  a  screw  at  any 
angle.  These  two  arms  are  applied  to  the  two  faces 
of  the  crystal  whose  interfacial  angle  is  to  be  meas- 
ured, and,  when  as  nearly  in  contact  as  possible,  the 
screw  is  set  and  the  angle  read  by  placing  the  arms 
upon  a  graduated  arc.  Such  measurements  are  not 
reliable  to  within  less  than  half  a  degree.  They  are 
chiefly  valuable  for  large  crystals  whose  faces  are 
rough  or  unpolished. 

For  accurate  measurements  of  the  interfacial  angles 
of  crystals  we  must  have  recourse  to  the  reflection- 
goniometer,  whose  construction  depends  upon  a  princi- 
ple first  made  use  of  by  Wollaston.  The  angle 
through  which  it  is  necessary  to  revolve  a  crystal 
about  one  of  its  edges,  so  as  to  successively  obtain  a 
reflection  of  the  same  object  from  the  two  planes 
whose  intersection  forms  the  edge,  is  equal  to  the  sup- 
plement of  their  interfacial 
angle. 

Suppose  (Fig.  24)  a  ray  of 
light  come  from  L,  and  be 
reflected  by  the  crystal 
plane  AO  to  the  eye  at  E. 
If  now  we  can  revolve  the 
crystal  about  the  edge  be- 
tween the  planes  AO  and 
FIG.  24.  CQ^  the  game  reflection  will 

be  sent  to  the  eye,  stationary  at  E,  from  the  plane 
(70,  when  it  has  been  brought  exactly  into  the  position 


GENERAL  PRINCIPLES.  23 

of  AO.  In  order  to  secure  this,  the  crystal  must  ob- 
viously be  revolved  through  the  angle  COc,  which  is 
the  supplement  of  AOC,  the  interfacial  angle  required. 
For  the  successful  application  of  this  principle  to 
practical  measurement  four  things  are  necessary : 

1.  The  eye  must  be  kept  at  the  same  point. 

2.  The  object  reflected  must  be  at  a  sufficient  dis- 
tance to  make  the  rays  coming  from  it  practically 
parallel. 

3.  The  edge  to  be  measured  must  be  parallel  to  the 
axis  of  revolution,  and  normal  to  the  plane  of  the 
graduated  circle  upon  which  the  angle  is  read.     (Ad- 
justment.) 

4.  This  edge  must  also  lie  exactly  in  the  continua- 
tion of  the  axis  of  revolution.     (Centering.)* 

2.   LAW  OF  THE  SIMPLE  MATHEMATICAL  RATIO. 
Crystallographic  Axes.    In  order  that  we  may  be  able  to 
classify  and  compare  the  forms  of  crystals,  we  must 
have  some  ready  and  simple  mode  of  expressing  the 

*  The  description  of  the  various  forms  of  reflection-goniometers, 
together  with  all  the  detail  of  their  construction  and  successful 
manipulation,  lies  far  beyond  the  scope  of  this  book.  For  informa- 
tion on  this  subject,  the  student  may  consult: 

E.  S.  Dana,  Text-book  of  Mineralogy;  2d  ed.,  1883;  pp.  83-88 
and  115-118. 

P.  Groth,  Physikalische  Krystallographie;  3d  ed.,  1885;  pp.  560- 
584. 

A  good  reflection-goniometer  is  quite  indispensable  to  all  who 
intend  to  do  any  special  work  in  crystallography.  The  best,  with 
horizontal  circles,  are  manufactured  by  R.  Fuess,  108  Alte  Jakob- 
strasse,  Berlin,  at  prices  ranging  from  $80  to  $350,  according  to  the 
completeness  of  their  equipment.  The  most  serviceable  for  all 
ordinary  purposes  is  his  Model  II,  costing  about  $150.  A  much 
simpler  instrument  with  vertical  circle,  constructed  on  the  old  Wol- 
laston  plan,  is  made  by  Voigt  &  Hochgesang  of  Gottingen. 


CRYSTALLOGRAPHY. 


+c 


-b- 


relative  positions  and  inclinations  of  their  planes.  This 
is  accomplished  by  referring  them  to  systems  of  axes, 
according  to  the  method  of  analytical  geometry. 

The  position  of  any  crystal  plane  is  thus  fixed  by  and 
expressed  in  the  relative  lengths  of  its  intercepts  on 
the  axes  to  which  it  is  referred.  The  axes  to  which  the 
planes  of  a  crystal  are  referred,  called  the  crystallo- 
graphic axes,  may  be  of  equal  or  unequal  length,  and 
may  intersect  at  either  oblique  or  right  angles. 

When   the   crystallographic   axes   are   of    unequal 

length,  it  is  customary  to 
designate  the  one  which 
stands  vertically  by  the 
letter  c,  the  one  which  runs 
from  right  to  left  by  6, 
+&  and  the  one  which  runs 
from  front  to  back  by  a. 
The  two  extremities  of  each 
axis  are  distinguished  by 
the  plus  or  minus  sign,  as 
shown  in  Fig.  25.  If  all 
three  axes  are  of  equal  length,  they  are  all  repre- 
sented by  a.  If  two  are  of  equal 
length,  they  are  designated  by 
a,  and  the  third  one  by  c. 

If  the  axial  intersections  are 
not  rectangular,  they  are  desig- 
nated by  the  Greek  letters  a, 
ft,  and  y,  as  follows  :  b  A  c  =  a, 
a  A  c  =  fi,  and  a  A  b  =  y.  (Fig. 
26.) 

The  planes  in  which  two  of  the 
crystallographic  axes  lie  are  called  axial  or  diametral 


— c 
FIG.  25. 


FIG.  26. 


GENERAL  PRINCIPLES. 


planes.  They  are  the  coordinate  planes  of  analytical 
geometry,  and  divide  the  space  within  the  crystal  into 
eight  seSfonts,  called  octants  ;  or,  in  one  system  where 
four  axes  are  used,  into  twelve  slants,  called  dodecants. 
The  axial  planes  are  also  sometimes  called  principal 
sections  (German,  Hauptschnitte). 

Parameters.  The  values  of  the  intercepts  of  any 
crystal  plane  on  the  axes  are  called  the  parameters  of 
the  plane.  They  are  expressed  in  terms  of  certain 
axial  lengths  which  are  assumed  as  unity.  Sup- 
pose (Fig.  27)  that  ABC  is  a  plane  which  intersects 
the  axes  X,  Y,  Z 
at  their  unit  lengths. 
The  position  of  any 
other  plane,  HKL, 
is  determined,  if  we 
know  the  values  OH, 
OK,  and  OL  in  terms 
of  OA,  OB,  and  OC. 

_    .,  .  OH          x 

In  this  case  ^y-,  =  2 , 

yr-^-  =  1 ;  -fr-^  =  ^.  These  quotients  are  the  parame- 
\J  Jj  \J  O  .  2i 

ters  of  the  plane  HKL. 

If  we  denote  the  axes  X,  Y,  Z  by  a,  b,  and  c,  the  most 
general  symbol  for  any  plane  becomes 

na  :  pb  :  me, 

where  n,  p,  and  m  are  rational  quantities  and  the 
parameters  of  the  plane. 

Since,  however,  any  plane  may  be  thought    of  as 
shifted  in  either  direction,  so  long  as  the  relative  value 


26  CRYSTALLOGRAPHY. 

of  its  intercepts  remains  the  same  (p.  14),  one  of  the 
three  parameters  may  always  be  made  equal  to  unity, 
and  the  most  general  expression  for  any  crystal  plane 
becomes 

na  :  b  :  me. 

Indices.  The  position  of  a  crystal  plane  may  be  equally 
well  expressed  by  employing  the  reciprocals  of  the 
parameters,  which  are  called  the  indices.  For  purposes 
of  notation  these  possess  many  practical  advantages 
over  the  parameters,  and  are  therefore  quite  generally 
used.  We  can  readily  see  from  Fig.  27  that  the  plane 

OA 
HKL    may    be    located    by   the    values    h  = 


OB  OC 

K  •=  -Tyrr'y  I  =  QT>  where  h,  K,  I  are  the  reciprocals  of 


n,  p,  ra,  the  parameters. 

Statement  of  the  Law.  With  the  aid  of  these  pre- 
liminary conceptions  of  what  is  meant  by  crystallo- 
graphic  axes,  parameters,  and  indices,  we  may  for- 
mulate the  law  of  the  simple  mathematical  ratio  (also 
known  as  the  law  of  the  rationality  of  the  indices)  as 
follows:  Experience  shows  that  only  those  planes  occur 
on  any  crystal,  whose  axial  intercepts  are  either  infinite  or 
small,  even  multiples  of  unity.  The  ratio  of  intercepts 
on  the  same  axis  for  all  planes  possible  on  the  same 
crystal  is  therefore  rational,  and  may  be  expressed  by 
small  whole  numbers,  simple  fractions,  infinity  or 


*  In  certain  cases  disturbances  during  the  growth  of  a  crystal,  or 
other  causes  not  understood,  produce  surfaces  whose  intercepts  are 
very  large,  if  not  almost  irrational.  Websky  has  called  such  faces 
vicinal  planes.  Their  significance  is  not  entirely  clear,  and  their 
consideration  lies  outside  of  the  scope  of  this  work. 


GENERAL  PRINCIPLES.  27 

This  law  leads  us  to  the  same  result  as  was  reached 
by  the  preceding  law  of  constant  interfacial  angles, 
viz.,  that  every  plane  cannot  occur  on  a  given  crystal. 
It  brings  us,  however,  the  additional  information  that 
those  planes  which  can  occur  must  have  rational  axial 
intercepts.  This  law,  like  its  predecessor,  is  a  neces- 
sary deduction  from  a  regular  molecular  structure, 
since  it  can  be  mathematically  demonstrated  that  only 
those  planes  which  possess  rational  indices  satisfy  the 
conditions  of  a  possible  crystal  plane  (p.  11.)  Thus 
observation  corroborates  our  hypothesis  of  molecular 
structure. 

Systems  of  Crystallographic  Notation.  In  orde'r  to 
compare  crystal  planes,  we  must  be  able  to  designate 
them  by  generally  applicable  systems  of  symbols. 
Many  such  systems  have  been  devised,*  but  those 
now  most  generally  employed  aim  to  locate  the  posi- 
tion of  each  crystal  plane  with  reference  to  the  crystal- 
lographic  axes,  and  are  therefore  based  upon  the  use 
of  either  parameters  or  indices. 

Parameter  System  of  Weiss,  This  is  one  of  the  oldest 
as  well  as  one  of  the  most  easily  understood  of  all 
systems  of  crystallographic  notation.  The  three  axes 
are  written  in  the  fixed  order  explained  on  page  24 : 
a  :  b  :  c  if  of  unequal  length ;  a  :  a  :  c  if  two  (then 
made  the  lateral  axes)  are  of  equal  length ;  or,  a  :  a  :  a 
if  all  three  are  of  equal  length.  To  each  axial  letter 
is  then  prefixed  the  numerical  value  of  its  parameter, 


*  Those  desiring  a  full  description  and  comparison  of  all  the  dif- 
ferent systems  of  crystallographic  notation  which  have  been  sug- 
gested, will  find  it  in  Goldschmidt's  Index  der  Krystallformen, 
vol.  i. 


'/f 

HT5SIV 


CRYSTALLOGRAPHY. 

whenever  this  is  not  unity.  In  the  most  general 
symbol  for  any  plane,  na  :  pb  :  me,  it  is  customary 
to  reduce  the  value  of  one  of  the  two  lateral  axes,  a 
or  5,  to  unity  (p.  26) ;  the  parameter  of  c  does  not, 
however,  become  unity  unless  the  parameter  of  one  of 
the  lateral  axes  is  at  the  same  time  unity.  For  this 
reason,  the  most  general  symbol  for  any  plane  in  the 
notation  of  Weiss  is  na  :  b  :  me  or  a  :  nb  :  me;  but  not 
na  :  mb  :  c.  Hence  the  value  of  the  parameter  n  can 
vary  only  between  one  and  infinity,  because  which- 
ever of  the  lateral  axes  (a  or  b)  is  the  shorter  is  as- 
sumed as  unity  ;  m,  on  the  other  hand,  varies  between 
zero  and  infinity,  because  it  refers  to  the  single  axis  c. 
For  example,  a  plane  whose  axial  intercepts  were  £  on 
a,  ^  on  b,  and  1  on  c,  would  not  be  written  \a  :  ^b  :  c, 
but  fa.  :  b  :  3c ;  so  again,  a  plane  whose  intercepts 
were  1  on  a,  J  on  b,  and  J  on  c  would  be  written, 
2a  :  b  :  fo,  etc.  If  all  three  axes  are  of  equal  length, 
and  therefore  interchangeable,  none  of  the  parameters 
ever  becomes  less  than  unity. 

Parallelism  of  any  plane  to  a  crystallographic  axis 
is  indicated  by  Weiss  with  the  sign  of  infinity,  oo , 
written  in  its  proper  place  as  a  parameter.  Thus  the 
symbol  of  a  plane  parallel  to  the  a  axis,  but  cutting  b 
and  c  at  unity,  becomes  oo  a  :  b  :  c ;  the  symbol  of  a 
plane  parallel  to  c  and  one  of  the  lateral  axes,  becomes 
GO  a  :  5  :  oo  c  or  a  :  oo  b  :  oo  c.  Since,  however,  one  of 
the  two  lateral  axes  must  always  be  unity,  the  symbol 
for  a  plane  parallel  to  both  lateral  axes-  is  not  written 
oo  a  :  oo  b  :  c,  but  a  :  b  :  Oc. 

This  notation  is  employed  in  the  writings  of  Weiss, 
Hose,  Quendstedt,  Rarnrnelsberg,  and  some  other  au- 
thors ;  but,  on  account  of  its  cumbersome  character, 


GENERAL  PRINCIPLES.  29 

it  has  generally  given  place  to  shortened  forms,  sug- 
gested by  Naumann  and  Dana. 

Abbreviated  Parameter  Symbols.  Naumann  has  pro- 
posed a  system  of  symbols,  now  in  quite  general  use, 
which  contain  in  their  centre  certain  capital  initials 
—  0  (octahedron)  when  the  planes  are  referred  to  a 
set  of  equal  axes,  and  P  (pyramid),  when  they  are  re- 
ferred to  systems  of  unequal  axes. 

Before  this  letter  is  written  the  parameter  m,  which 
refers  to  the  vertical  axis,  and  after  it  the  parameter 
n,  which  belongs  to  one  of  the  lateral  axes.  Any 
parameter  whose  value  is  unity  is  omitted.  Thus 
the  most  general  symbol  becomes  mPn.  P  or  0 
signifies  a  form  whose  planes  cut  all  three  axes  at 
unity;  raP,  a  form  whose  lateral  axes  are  both  unity ; 
and  Pn,  a  form  whose  vertical  axis  is  unity  and  one 
of  whose  lateral  axes  is  n.  Parallelism  to  any  axis 
is  represented  by  the  sign  of  infinity,  GO  ,  in  its 
proper  place  in  the  symbol,  co  P  oo  signifies  a  plane 
parallel  to  the  vertical  and  one  lateral  axis ;  but  the 
symbol  Pcoyco  (oo  a  :  oo  b  :  c)  is  replaced  by  its  equiva- 
lent, OP  (a  :  b  :  Oc),  for  a  plane  parallel  to  both  lateral 
axes,  as  in  the  notation  of  Weiss.  Many  other  points 
regarding  the  notations  of  Weiss  and  Naumann  will 
be  brought  out  in  the  course  of  the  descriptions  of 
the  forms  of  the  various  crystal  systems. 

J.  D.  Dana  has  suggested  a  further  simplification  of 
Naumann's  symbols,  which  has  come  into  current 
use  in  the  United  States.  It  consists  in  substituting 
a  hyphen  for  Naumann's  initial,  and  the  letter  i  or  7  for 
the  sign  of  infinity.  The  fundamental  form  is  repre- 
sented by  1 ;  a  single  parameter  is  written  alone,  if  it 
refers  to  the  vertical  axis,  c ;  but  it  is  preceded  by  1- 


if  it  refers  to  one  of  the  lateral  axes.  Thus  Naiinmnn's 
symbol  *1()*1  or  kJ/*'J  becomes,  in  Dana's, notation,  "l-'l ; 
2P,  2 ;  P3,  13  ;  P,  1 ;  oo  P,  /;  P  oo,i^o  ;  oo  P  oo,  t-t ;  / 
OP,  0;  8Pf,  8-f;  etc.  These  symbols  can  pn >s« -ni 
no  diilicnlty  to  anyone  familiar  with  those  of  Nan- 
inann. 

Index  System  of  Miller.  A  system  of  cr\  -si  allograph  ic 
n.d;ifi«ii  h.Msrd  upon  iln^  us(^  of  tlu>  indices  owes  ils 
pn*s(Mii  wide  application  to  the  writings  of  Prof.  \V.  Jl. 
Miller.  Hence  it,  is  called  the  IMiller  system,  although 
il  was  lirsl  devised  in  1825  by  Whewell. 

In  this  svsieni  the  synihol  of  any  plane  consists  of 
(.lie  reciprocals  of  I  lie  parameters,  writ  ten  in  the  order 
a. hove  £\  ven  for  t  he  axes  <i.  A,  c,  a  ml  in  the  simplest  form 
possible  \\hen  they  a.re  cleared  of  fractions.  This  can 
he  made  most  clear  hy  an  example.  The  symbol  M/'^ 

of  Nftiimann  becomes  4ct  i  b  i  3c  in  the  notation  of 
Weiss.  The-  reciprocals  of  these  parameters  are  jj, 
I,  ',,  which,  when  cleared  of  fractions,  become  !2  ,'{  1 
(read  tiro,  thm\  mir).  This  is  the  symbol  for  the 
same  plane  in  the  notation  of  Miller.  The  indices  are 
always  written  in  the  same  order,  and  without  any 
symbol  for  the  axes  or  crystal  system. 

A  plane  parallel  to  an  axis  must,  of  course,  contain 
the  index  0  (the  reciprocal  of  infinity)  in  its  symbol. 
Thus  oo  P  oo  becomes  100  or  010 ;  OP,  001 ;  oo  P,  110; 
Poo ,  Oil  or  101;  eta* 

*  A  modification  of  the  use  of  indices  as  crystal  symbols  has  re- 

;i;>  l.rni   surest. •(!   hy  (Jol.Isclunidl.      Thr   Miller  syml.,>l 
i     li..i  i,  nod  from  three  terms  to  two  by  reducing  the  third  to  unity. 

( ;<  .Idachmidt's  indices,  p  and  g,  are  =  j  and  y.    In  this  case  it  is  of 

roursr  impossible  to  avoid  fr:u  I  ion  ,  -r,  Indrx  drr  Kryslallfornu-n 
dn  Miii. TMlini.  vol.  i.  p.  TJ). 


81 

1 1,  is  evident,  that  the  symbols  of  Naumann  ;i,n<l 
Daua  stand,  nod  for  single  planes,  hut  for  ;ill  the 

planes  which  have  equal  intercepts  on  equivalent 
axes.  Such  an  assemblage  of  planes  is  technically 
known  ;is  u  crystal  form  (see  beyond,  p.  ,'{.r>).  The 
symbols  of  Weiss  distinguish  between  those  pianos 
of  ;i,  form  which  belong  to  the  same  oci;i,nt,  by  vary- 
ing the  order  in  wliich  their  |>;i,r;unefers  me  written  ; 

they  do  not,  however,  ordinarily  locate  a  plane  in 
any  partieular  octant,  although  this  may  be  done 
by  Higim. 

It  is  an  advantage  to  lie  a,ble,  to  desi^n;it(^  ;i,ny  |>;i,r- 
tir.iilar  phi.iie  of  a  crystal  form,  ;ind  this  is  done  in  the 
notation  of  Miller  by  the  use  of  si^ns.  Any  index  re- 
ferring to  a  negative  <!iid  of  an  axis  (MM  Fig.  25)  has 
a  minus  sign  written  over  it.  This  serves  to  locate  a 
given  plane  in  ono  particular  octant,  thus : 

hid  JJd  lid  lU 

JiTd  hid  IE  M 

The  members  of  each  pair. of  parallel  planes  have 

the  same  indices  and  complementary  signs;  hence  to 
change  the  signs  of  any  Miller  symbol  is  to  oliange 
the  plane,  to  its  parallel  and  therefore  equivalent  piano 
on  the  opposite  side  of  the,  crystal. 

Miller  writes  the  indices  of  a  single  plane  either 
alone,  /////,  or  inclosed  in  parenthesis,  (It/cl).  If  an  en- 
tire, form  is  to  lie  represented,  the  indices  are  inclosed 
in  brackets,  \Uld\* 

*  It  is  <|uil<-  « •••...< -nli.-il  Hint,  Hie  student,  .should  brroinr  equally  fu- 
iniliur  wil.li  tin-  symboln  of  Weiss,  Nuiiinann  (or  Dunn),  and  Miller. 
Kor  this  pin  | »«.:.«•  |.I:L( -i.ii •<•  in  t.ruiiHforiniiiK  •!><;  Hyiubols  of  one  sys- 


32  CRYSTALLOGRAPHY. 

Levy's  System,  The  oldest  system  of  crystallographic 
notation  was  devised  by  Haiiy  and  subsequently  modi- 
fied by  Levy  and  Des  Cloizeaux.  It  is  still  in  gen- 
eral use  in  France,  but,  on  account  of  its  complicated 
and  difficult  character,  it  has  no  currency  in  other 
countries.  No  explanation  of  this  system  need  be  at- 
tempted here.  Further  information  regarding  it  may 
be  found,  if  desired,  in  the  works  of  Des  Cloizeaux, 
Goldschmidt,  and  Groth. 

3.    LAW  OF  SYMMETRY. 

Statement  of  the  Law.  It  has  been  found  that  an  im- 
portant property  of  crystal  forms,  and  one  according 
to  which  they  may  be  advantageously  classified,  is 
their  symmetry. 

A  plane  of  symmetry  may  be  defined  as  a  plane  which 
is  capable  of  dividing  a  body  into  two  halves  which 
are  related  to  each  other  in  the  same  way  that  an 

tern  into  those  of  the  others  is  recommended  until  the  matter  pre- 
sents no  further  difficulty.  A  few  examples  for  such  practice  are 
here  appended: 

Weiss.                                 Naumann.            Dana.  Miller. 

a:      a:      a                         0                    1  Jill} 

^:      aropa                         oo  0                  1  (Ott)  (?/<)) 

oo  a:  oo  a:       a                      ooOoo                i-i  (001) 

a :      a  :       c  or  a :  b  :c         P                   1  {ill} 


oo  a  :      b  :  oo  c 
a  :  ooi  :  ooc 


2a  :  6  :  2c 

3a:  b:  c 

a:  b:  3c 

a :  26 :  3c 

f  a  :  6  :  3c 


ooPoo 

£*              1  010  1 

ooPob  ' 

i4 

100] 

Poo 

\-i 

no  I 

2P2 

2-2 

121} 

P3 

1-3 

133} 

3P 

3 

331  ! 

3P2 

3-2             {632 

3Pf 

3-|             {231 

GENERAL  PRINCIPLES. 


33 


object  is  to  its  reflection  in  a  mirror.*  The  grade  of 
symmetry  which  any  crystal  form  possesses  is  con- 
ditioned by  the  number  of  its  planes  of  symmetry. 
For  example,  Figs.  28  and  29  represent  two  crystals, 
the  one  possessed  of  five  planes  of  symmetry,  the  other 
of  but  one. 


FIG.  28. 


FIG.  29. 


The  law  of  symmetry  may  be  formulated  as  follows  : 
All  the  faces  of  a  crystal  are  grouped  in  accordance  with 
certain  definite  planes  of  symmetry  which  are  fixed  in  their 
position  for  the  same  crystal,  and  condition,  not  merely  its 
external  form,  but,  in  an  equal  degree,  the  distribution  of 
all  of  its  internal  physical  properties. 

This  law  shows  that  symmetry  is  primarily  a  prop- 
erty of  the  internal  molecular  structure  of  crystals, 
and  that,  on  this  account,  it  is  expressed  in  their  out- 
ward form.  This  is  the  true  explanation  of  its  im- 
portance to  crystallography. 

In  order  to  bring  out  clearly  the  symmetry  of  any 

*  More  exactly  we  may  say:  two  objects,  or  two  halves  of  the 
same  object,  are  symmetrical  with  reference  to  a  plane  placed  be- 
tween them,  when  from  any  point  of  one  object  a  normal  to  this 
plane,  prolonged  by  its  own  length  on  the  opposite  side  of  the  plane, 
will  meet  the  corresponding  point  of  the  other  object. 


34 


CRYSTALLOGRAPHY. 


crystal  form,  we  must  imagine  it  freed  from  all  distor- 
tion and  restored  to  its  ideal  proportions  (see  p.  14). 
Planes  of  symmetry  are  of  two  kinds :  those  which 
contain  two  or  more  equivalent  and  interchangeable 
directions,  and  those  which  have  no  such  directions. 
The  first  are  called  principal,  and  the  others  secondary 
planes  of  symmetry.  This  difference  can  be  best  illus- 
trated in  a  concrete  case.  Fig.  30  represents  a  form 
two  of  whose  axes,  a  — a  and 
a'  — a',  are  of  equal  length,  while 
the  third  is  of  unequal  length. 
According  to  our  definition,  the 
axial  planes,  aa'aa'  and  acac,  are 
both  planes  of  symmetry ;  but,  in 
the  first,  the  directions  a  — a  and 
a'  — a'  are  equivalent  and  inter- 
changeable, while,  in  the  second 
plane,  the  directions  a  — a,  c  - — c 
are  not  interchangeable. 

Again,  the  crystal  form  (Fig.  30) 

can  be  brought  exactly  into  its  original  position  by  a 
revolution  of  less  than  180°  about  the  axis  c ;  hence  c 
is  a  principal  axis  of  symmetry.  It  is,  however,  im- 
possible to  bring  it  into  its  original  position  by  revolv- 
ing it  less  than  180°  about  a  — a  or  a!  — a! ;  hence 
these  are  called  secondary  axes  of  symmetry.  Principal 
axes  of  symmetry  are  always  normal  to  principal 
planes  of  symmetry,  and  secondary  axes  of  symmetry, 
to  secondary  planes  of  symmetry. 

Axes  of  symmetry  are  always  chosen  as  the  crystal- 
lographic  axes,  whenever  there  are  three  or  more  of 
them  present.  Moreover,  when  they  are  present, 
principal  axes  of  symmetry  are  always  preferred  for 


FIG.  30. 


GENERAL  PRINCIPLES. 


35 


this  purpose  to  secondary  axes.  If  only  one  principal 
axis  is  present,  then  two  secondary  axes  are  taken  in 
addition.  If  only  one  axis  of  symmetry  of  any  kind  is 
present,  then  two  other  arbitrary  directions  are  se- 
lected as  crystallographic  axes. 

The  Crystal  Form.  Thus  far  the  term/orm  has  been 
employed  in  a  somewhat  loose  sense.  It  has,  however, 
in  crystallography  a  very  particular  and  technical 
meaning,  which  may  be  defined  as  the  sum  of  those 
planes  tvhose  presence  is  required  by  the  symmetry  of  the 
crystal,  when  one  of  them  is  present.  In  other  words, 
a  crystal  form  embraces  all  those  planes  which,  irre- 
spective of  signs,  can  be  repre- 
sented  by  a  single  symbol.  We 
shall  in  future  employ  the 
word/orm  only  in  this  special 
sense. 

The  number  of  planes  com- 
posing a  form  depends  upon 
the  grade  of  symmetry  and  in- 
creases as  the  symmetry  in- 
creases. For  example,  if  a  crys- 
tal is  entirely  without  planes 

of  symmetry,  then  the  presence  of  any  plane,  A  (Fig. 
31),  necessitates  the  occurrence  of  only  its  parallel 
plane,  A'.  In  this  case,  therefore,  a  complete  crystal 
form  is  composed  of  only  two  planes. 

If,  however,  a  single  plane  of  symmetry  ( WXYZ)  is 
present  (Fig.  32),  then  each  of  the  two  planes,  A  and 
A',  necessitates  the  occurrence  of  another  plane  (B 
and  B')  symmetrical  to  it.  In  this  case  the  complete 
form  consists  of  four  similar  planes. 

Thus   it   can  be  seen  that  the  number  of    planes 


36 


CRYSTALLOGRAPHY. 


W 


Fio.  32. 


which  go  to  make  up  a  complete  crystal  form  depends, 
in  a  way,  upon  the  grade  of  sym- 
metry of  this  form. 

Costal  forms  are  divided  into 
three  types,  according  as  their  planes 
intersect  one,  two,  or  three  of  the 
axes  of  reference. 

Pinacoids  (nivaZ,  a  board)  are 
composed  of  planes  parallel  to  two 
axes.  They  correspond  in  position 
to  the  faces  of  the  cube. 

Prisms  are  forms  whose  planes 
intersect  two  axes,  while  they  are 
parallel  to  the  third.  When  such  forms  are  paral- 
lel to  either  of  the  two  lateral  axes,  they  are  called 
domes. 

Pyramids  are  forms  whose  planes  cut  all  three  axes. 
Crystal  forms  are  furthermore  dosed  when  their 
planes  completely  enclose  space,  and  open  when  they 
do  not.  Those  forms  whose  planes  are  all  parallel 
to  a  single  line  are  open  forms.  They  cannot,  of 
course,  occur  alone,  but  only  in  combination  with  other 
forms  of  the  same  grade  of  symmetry. 

Crystal  Combinations.  Inasmuch  as  certain  crystal 
forms  do  not  by  themselves  enclose  space,  they  cannot 
occur  alone,  and  it  is  equally  true  that  the  occurrence 
alone  of  closed  forms  is  rather  the  exception  than  the 
rule.  The  simultaneous  occurrence  on  a  single  in- 
dividual of  two  or  more  crystal  forms  is  technically 
known  as  a  combination. 

The  following  points  are  of  particular  importance 
in  regard  to  crystal  combinations  : 

1.  Only  forms  possessing   the  same  grade  of  sym- 


GENERAL  PRINCIPLES. 


37 


rgfiirj-_can  combine.  This  is  evident  if  we  remember 
that  the  external  form  of  a  crystal  is  only  an  outward 
expression  of  its  molecular  structure ;  and  that  this 
must,  of  course,  be  the  same  throughout  an  entire 
individual. 

2.  When  two  or  more  forms  combine,  the  axes  of  all 
must  be  coincident  and  possess  the  same  relative,  but 
not  the  same  absolute  lengths.  This  point  is  apt  to  pre- 
sent difficulties  to  a  beginner.  It  must  be  remembered 
that  it  is  the  relative  and  not  the  absolute  lengths 
of  the  axes  which  are  essential  in  determining  the  po- 
sition of  any  plane.  A  cube  and 
octahedron,  for  example,  whose 
axes  are  of  equal  length  could 
not  possibly  combine,  because 
the  cube  would  entirely  enclose 
the  octahedron  (Fig.  33).  In  or- 
der that  these  two  forms  may 
combine,  the  axes  of  the  cube 
must  be  relatively  the  shorter.  If  they  are  only 
slightly  so,  the  cube  will  appear  as  square  truncations 
on  the  octahedral  angles  (Fig.  34).  If,  however,  the 
cubic  axes  are  much  the  shorter,  the  two  forms  will 
appear  in  equal  development  (Fig.  35),  or  the  octa- 
hedron may  form  triangular  truncations  on  the  cubic 
angles  (Fig.  36). 

The  relative  development  of  the  planes  of  different 
forms  occurring  in  combination  is  an  important  factor 
in  conditioning  the  habit  of  a  crystal.  This  is  also 
largely  influenced  by  distortion  and  elongation,  as  ex- 
plained on  p.  14. 

Certain  terms  are  employed  in  describing  the  re- 
placement of  crystal  edges  and  angles  by  the  planes  of 


FIG.  33. 


38 


CRYSTALLOGRAPHY. 


other  forms,  which  here  need  definition.  The  replace- 
ment of  an  edge  or  angle  by  a  single  plane  is  called  a 
truncation  (German,  Abstumpfung} ;  a  replacement  by 
two  planes  is  called  a  bevelment  (German,  Zuschdrfung). 


FIG.  34. 


FIG.  35. 


FIG.  36. 


When  a  solid  angle  is  replaced  by  more  than  two 
planes  it  is  said  to  be  acuminated  or  blunted  (German, 
Zuspitzung).  If  a  truncating  plane  makes  equal  angles 
with  the  planes  on  each  side  of  it,  it  is  said  to  be 
symmetrical  (German,  gerade  Abstumpfung).  If  such 
angles  are  unequal,  the  truncation  is  unsymmetrical  or 
oblique  (German,  schiefe  Abstumpfung).  Of  the  following 
figures,  37  and  38  are  examples  of  symmetrical  trunca- 


FIG.  37. 


FIG.  38. 


FIG.  39. 


tions,  the  first  of  edges,  the  second  of  angles.  Fig.  39 
shows  an  unsymmetrical  truncation  of  edges,  while 
Figs.  40  and  41  represent  bevelments. 


GENERAL  PRINCIPLES. 


39 


It  is  a  convenient  fact  to  remember  that  the  indices 
of  any  plane  which  symmetrically  truncates  an  edge 
between  two  planes  of  the  same  form  may  be  found  by 


FIG.  41. 


taking  the  algebraic  sum  of  the  indices  of  the  two 
planes  forming  the  edge  truncated.  Thus  an  edge  be- 
tween the  planes  111  and  111  would  be  symmetrically 
truncated  by  the  plane  110,  an  edge  between  321  and 
312  by  633  or  211,  etc. 

Independent  Occurrence  of  Partial  Crystal  Forms.  By  the 
distortion  of  crystal  forms  due  to  irregular  growth 
(p.  14),  certain  planes  may  be  reduced  in  size  to  mere 
points  and  so  disappear.  Such  an  irregular  absence 
of  one  or  more  planes  belonging  to  a  complete  form 
is  purely  accidental  and  of  no  particular  significance. 
It  is  known  as  merohedrism  (ftepo?,  a  part,  and  edpa, 
face). 

Hemihedrism  and  Tetartohedrism.  In  many  other 
cases,  however,  crystal  forms  do  occur,  which,  if  we 
adhere  to  the  definition  of  the  complete  form  given  on 
p.  35,  must  be  regarded  as  partial.  Such  partial 
forms,  on  account  of  the  regularity  of  their  develop- 
ment and  the  frequency  of  their  occurrence,  are  of  the 
highest  importance  in  crystallography;  and  to  them 
must  be  accorded  as  full  recognition  as  to  those  which 


40  CRYSTALLOGRAPHY. 

fulfil  all  the  requirements  of  symmetry.  Their  exist- 
ence is  best  explained  on  the  assumption  that  the 
planes  composing  one  half  or  one  quarter  of  certain  com- 
plete forms  are  capable  of  occurrence,  entirely  inde- 
pendent of  their  other  halves  or  quarters. 

The  theoretical  consideration  of  all  possible  regular 
arrangements  of  points  in  space  which  satisfy  the  con- 
ditions of  crystal  structure  (p.  6)  shows  that  these 
also  include  partial  forms,  similar  to  those  observed  in 
nature. 

In  contrast  to  such  partial  forms,  the  completely 
symmetrical  crystal  is  termed  holohedral  (oXos,  whole, 
and  'edpa,  face),  or  a  holohedron ;  while  the  half  forms 
are  called  hemihedral  (rf^i,  half,  and  edpa,  face)  or 
hemihedrons,  and  the  quarter  forms  are  known  as 
tetartohedral  (rhapTo?,  quarter,  and  'sdpa,  face)  or 
tetartohedrons.  % 

We  may  imagine  any  hemihedral  or  tetartohedral 
form  as  produced  by  the  suppression  of  a  certain  half 
or  three  quarters  of  the  planes  com- 
posing the  complete  or  holohedral 
form,  and  the  extension  of  those 
planes  which  remain  until  they  meet. 
Let  us,  for  instance,  suppose  that 
the  white  planes  of  the  inside  figure 
(Fig.  42)  are  suppressed,  and  that 
the  shaded  planes  are  extended  to 
intersection ;  they  will  then  pro- 
FIG.  42.  duce  the  outside  figure,  which  is  the 

hemihedron  corresponding  to  the  interior  holohedron. 
Hemihedral  and  tetartohedral  forms  cannot,  how- 
ever, be  produced  by  the  selection  and  extension  of 
any  arbitrary  half  of  the  planes  belonging  to  the  cor- 


GENERAL  PRINCIPLES.  41 

responding  holohedral  form.  On  the  contrary,  the 
planes  capable  of  producing  partial  forms  must  be 
selected  in  accordance  with  certain  definite  conditions. 
These  may  be  stated  as  follows :  If  the  crystal  be  im- 
agined as  free  from  all  distortion,  then  only  such  planes  of 
the  complete  form  can  survive  to  produce  a  hemihedfon  or 
tetartohedron  as  uoill,  after  their  extension,  intersect  the 
extremities  of  all  equivalent  axes  of  symmetry  in  the  same 
number,  under  equal  angles  and  at  the  same  distance  from 
the  centre. 
Figs.  43  and  44  show  how  the  halves  of  the  planes 


FIG.  43.  Fio.  44. 

of  a  holohedron  may  be  selected  so  as  in  the  first 
instance  to  fulfil,  and  in  the  second  not  to  fulfil,  the 
above  conditions.  For  in  the  first  case  the  vertical 
axis  is  cut  at  its  extremities  by  pairs  of  shaded  planes, 
just  as  the  lateral  axes  are ;  while,  in  the  second  case, 
the  vertical  axis  is  still  cut  by  pairs  of  shaded  planes, 
but  the  lateral  axes  by  single  ones,  which  alternate 
with  white  planes. 

Every  complete  crystal  form  is  bounded  by  pairs  of 
parallel  planes  (p.  18).  Such  a  form  may  become 
hemihedral  (1)  by  losing  one  half  of  its  pairs  of 
planes,  the  other  pairs  remaining  intact ;  or  (2)  by 
losing  one  plane  from  each  of  its  pairs,  so  that  no 


42  CRYSTALLOGRAPHY. 

plane  of  the  resulting  hemihedron  has  another  paral- 
lel to  it.  The  first  method  produces  what  is  called 
parotid-face  hemihedrism  (Fig.  45),  and  the  second  what 
is  known  as  indined-face  hemihedrism  (Fig.  46). 


FIG.  45.  Fio.  46. 

Inasmuch  as  all  crystal  forms  are  the  external 
expressions  of  an  internal  molecular  structure,  the 
law  of  combinations  (p.  36)  must  hold  good  not  merely 
for  holohedral,  but  also  for  hemihedral  and  tetarto- 
hedral  crystals.  Only  forms  belonging  to  the  same 
kind  of  hemihedrism  or  tetartohedr-ism  can  combine. 
Apparent  exceptions  to  this  law  are  caused  by 
the  fact  that  certain  holohehral  forms  are  incapable 
of  producing  hemihedrons  which  are  geometrically 
different  from  themselves.  This  will  be  more  fully 
explained  in  the  succeeding  chapters. 

Hemimorphism.  One  half  of  the  planes  bounding  a 
holohedron  sometimes  occur  independently  of  the 
other  half,  where  the  selection  of  faces  cannot  be 
brought  within  the  above-given  conditions  of  hemi- 
hedrism. Instead  of  the  new  form  having  one  half  of 
the  planes  similarly  grouped  about  either  extremity 
of  an  axis  of  symmetry,  we  may  have  all  of  the  holo- 
hedral planes  at  ojie  extremity  of  such  an  axis,  and 
none  of  them  at  the  other.  This  mode  of  development 


GENEEAL  PRINCIPLES.  43 


is  called  hemimorphism  (rf^i,  half,  juop0?/,  form),  and  the 
axis  with  reference  to  which  the  planes  are  grouped 
is  known  as  the  hemimorphic  axis. 

Hemimorphic  forms  cannot  of  themselves  enclose 
space.  They  must,  therefore,  always  occur  in  combina- 
tion with  other  hemimorphic  forms,  as  is  shown  in  the 
case  of  calamine  (Fig.  47).  That  this 
development  is  the  direct  result  of 
molecular  structure  is  shown  both  by 
the  hemimorphic  indentations  pro- 
duced when  the  crystal  planes  are 
etched,  and  by  certain  peculiar  physi- 
cal properties,  most  prominent  among 
which  is  pyro-electricity.  When  hemi- 
morphic crystals  are  heated,  they 
give  off  positive  electricity  at  one  end  (analogue  pole), 
and  negative  at  the  other  (antilogue  pole}.  When  the 
temperature  begins  to  fall,  this  order  is  reversed. 

Symmetry  of  a  Crystal  Plane.  The  symmetry  of  a 
crystal  plane  is  conditioned  by  the  number  of  planes  of 
symmetry  to  which  it  is  normal.  Thus  a  face  perpen- 
dicular to  one  plane  of  symmetry  is  called  monosym- 
metric  ;  one  perpendicular  to  two  planes  of  symmetry, 
disymmetric;  etc. 

The  Crystal  Systems.  It  can  be  proved  in  a  variety  of 
ways  that  all  the  complete  or  holohedral  crystal  forms 
that  are  possible  must  possess  one  of  six  grades  of 
symmetry.  Symmetry,  therefore,  furnishes  a  valuable 
means  of  classifying  the  complete  forms  into  six 
groups,  called  Crystal  Systems.  From  the  symmetry 
of  the  holohedral  forms  the  crystallographic  axes,  to 
which  their  planes  are  referred,  are  deduced,  as  al- 
ready stated  on  p.  34, 


44  CRYSTALLOGRAPHY. 

A  definition  of  the  crystal  systems,  however,  which 
is  based  entirely  upon  symmetry  cannot  be  wholly 
satisfactory,  since  it  strictly  excludes  the  partial  crystal 
forms  (hemihedral,  tetartohedral,  and  hemimorphic), 
whose  grade  of  symmetry  is  always  less  than  that  of 
their  corresponding  holohedrons.  These  partial  forms 
are  nevertheless  referable  to  the  same  crystallographic 
axes  as  the  complete  forms  from  which  they  are  de- 
rived, and  hence  the  crystal  systems  must  be  defined 
in  terms  of  both  their  symmetry  and  axes :  A  system 
is  the  sum  of  aU  the  possible  crystal  forms  whose  planes 
can  be  referred  to  the  same  kind  of  axes ;  or  the  sum  of 
all  possible  HOLOHEDRAL  forms  which  possess  the  same 
grade  of  symmetry. 

According  to  this  definition,  we  may  characterize 
the  six  crystal  systems  as  follows  : 

CLASS  I.  (Isometric.)  1.  All  forms  referable  to  three 
axes  of  equal  length  which  intersect  at  angles  of  90°. 
Holohedral  forms  are  possessed  of  three  principal 
planes  of  symmetry  at  right  angles  to  one  another 
(giving  the  three  rectangular  axes) ;  and  six  secondary 
planes  of  symmetry,  which  bisect  each  of  the  angles 
between  the  principal  planes.  .  .  Isometric  System. 

CLASS  II.  (Isodimetric.)  All  forms  referable  to  one 
principal  or  vertical  axis,  which  is  perpendicular  to, 
and  different  in  length  from  the  lateral  axes.  One 
principal  plane  of  symmetry,  giving  the  principal  axis. 

2.  Number  of  equal  lateral  axes  two,  intersecting 
the  principal  axis  and  each  other  at  angles  of  90°. 
Number  of  secondary  planes  of  symmetry  for  holo- 
hedral  forms  four,  which  are  all  perpendicular  to  the 
principal  plane  of  symmetry,  and  inclined  to  each  other 
at  angles  of  45°,  90°,  and  135°.  .  Tetragonal  System. 


GENERAL  PRINCIPLES.  45 

3.  Number  of  equal  lateral  axes  three,  intersecting 
the  principal  axis  at  angles  of  90°,  and  each  other  at 
angles  of  60°.     Number  of  secondary  planes  of  sym- 
metry for  holohedral  forms  six,  all  at  right  angles  to 
the  principal  plane  of  symmetry,  and  inclined  to  each 
other  at  angles  of  30°,  60°,  90°,  120°,  and  150°. 

Hexagonal  System. 

CLASS  III.  (Anisometric.)  No  principal  axis  or  plane 
of  symmetry. 

4.  All  forms  referable  to  three  axes  of  unequal  length 
intersecting  at  right  angles.     Holohedral  forms  pos- 
sessed of  three  secondary  planes  of  symmetry  at  right 
angles. .      Orthorhombic  System. 

5.  All  forms  referable  to  three  axes  of  unequal  length, 
two  of  which  intersect  at  an  oblique  angle,  while  they 
are  both    perpendicular  to  the  third.      One  second- 
ary plane  of  symmetry.      .     .     .      Monoclinic  System. 

6.  All   forms  referable  to   three   axes  of  unequal 
length,  all  oblique  to  one  another.     No  plane  or  axis 
of  symmetry.     „ Triclinic  System. 

We  may  now  proceed  to  the  description  of  the  par- 
ticular types  of  crystal  forms  which  compose  each  of 
these  six  systems. 


CHAPTEE  III. 


THE  ISOMETRIC  SYSTEM,* 
HOLOHEDRAL  DIVISION. 

Symmetry.  The  special  characteristics  of  a  crystal 
system  may  be  advantageously  deduced  from  the 
symmetry  of  its  holohedral  forms,  for  this  belongs 
primarily  to  the  molecular  structure  of  the  crystals 
themselves. 

According  to  the  definition  of  the  isometric  system 
given  in  the  preceding  chapter,  its  complete  forms 
possess  the  highest  grade  of  symmetry  which  is  con- 
sistent with  the  law  of  rational  indices.  This  is 

nine-fold,  and  is  distributed 
about  three  principal  planes 
of  symmetry  at  right  angles 
to  one  another,  the  angles 
between  which  are  bisect- 
ed by  six  other  secondary 
planes  of  symmetry.  The 
position  of  these  planes  of 
symmetry  is  shown  in  the 
accompanying  figure  (No.  48),  where  the  three  princi- 
pal planes  are  shaded  and  indicated  by  Roman  nu- 
merals, while  the  secondary  planes  are  white,  and  are 
numbered  1,  3»  3,  4,  5,  6. 

The  symmetry  of  partial  "forms  in  the  isometric 
system  is  necessarily  less  than  that  of  the  complete 

*  Also  called  the  regular,  tesseral,  teasular,  or  cubic  system. 

46 


FIG.  48. 


THE  ISOMETEIC  SYSTEM.  47 

forms,  and  its  character  will  be  fully  explained  be- 
yond, as  each  of  the  hemihedral  and  tetartohedral 
divisions  is  considered. 

Axes.  A  still  more  comprehensive  definition  of  the 
isometric  system,  inasmuch  as  it  includes  both  partial 
and  complete  forms,  is  that  it  is  the  sum  of  all  crystal 
forms  whose  planes  are  referable  to  three  axes  of  equal 
length  ivhich  intersect  at  right  angles.  These  axes  are  di- 
rectly deducible  from  the  holohedral  isometric  sym- 
metry, because  they  represent  the  three  principal  axes 
of  symmetry;  and,  as  has  been  already  stated  (p.  34), 
principal  axes  of  symmetry  are  employed  as  crystal- 
lographic  axes  whenever  they  are  present.  Moreover, 
because  these  axes  lie,  two  and  two,  in  the  principal 
planes  of  symmetry,  they  must  be  not  merely  rectan- 
gular and  of  equal  lengtht  but  also  all  interchangeable;  i.e., 
whatever  is  true  of  one  must  be  true  also  of  the  other  two.* 

The  Fundamental  Form.  The 
starting-point  for  any  series  of 
planes  which  is  referable  to  the 
same  set  of  rectangular  crys- 
fcallographic  axes,  is  a  form  which 
cuts  all  of  the  axes  at  their 
unit  length.  This  is  called  the 
fundamental  or  ground-form  for 
the  series.  In  the  isometric  FlG* 49> 

system  all  of  the  axes  have  the  same  length,  and  its 

*  Two  other  sets  of  axes  are  of  use  in  the  isometric  system.  One  is 
the  set  of  intersection-lines  between  the  principal  and  secondary 
planes  of  symmetry;  and  the  other  the  intersection -lines  of  the 
secondary  planes  of  symmetry  with  each  other.  The  first  are  normal 
to  the  faces  of  the  rhombic  dodecahedron  and  are  called  the  digo- 
nal;  the  second  are  normal  to  the  faces  of  the  octahedron  and  are 
called  the  trigonal  axes. 


48  CRYSTALLOGRAPHY. 

ground-form  must  therefore  be  one  whose  planes  cut 
all  the  axes  at  the  same  distance  from  the  centre. 
This  is  the  regular  octahedron,  whose  eight  equilateral 
sides  intersect  at  angles  of  109°  28'  16.4"  (Fig.  49). 
The  parameters  and  indices  of  this  form  are  all  alike 
unity.  Modified  by  signs  indicating  particular  octants, 
as  explained  on  p.  31,  the  indices  of  the  octahedron  are : 
(Above)  111,  111,  111,  111. 

(Below)          111,  111,  111,  111. 

Derivation  of  the  Types  of  Holohedral  Forms  possible  in 
the  Isometric  System.  Before  attempting  to  describe 
the  isometric  crystal  forms,  it  will  be  well  to  discover 
how  many  types  of  such  forms  are  possible  ;  i.e.,  how 
many  different  kinds  of  forms  possess  the  complete 
isometric  symmetry  and  have  their  planes  referable  to 
three  equal  and  rectangular  axes.  To  do  this,  we  may 
first  find  what  is  the  most  general  form  consistent 
with  these  conditions,  and  then  deduce  from  this  all 
possible  special  cases,  by  giving  definite  or  limiting 
values  to  one  or  both  of  its  parameters. 

The  most  general  symbol  for  any  plane  referred  to 
a  set  of  three  perpendicular  axes,  expressed  in  the 
parameter  notation  of  Weiss,  is,  as  was  explained  on 
p.  26,  na  :  b  :  me.  Since,  however,  in  the  isometric 
system  all  three  axes  are  equal  and  interchangeable, 
they  are  represented  by  the  same  letter,  a,  and  this 
general  formula  therefore  becomes  na  :  a  :  ma.  . 

Of  this  symbol,  as  it  stands,  six,  and  only  six,  permu- 
tations are  possible,  as  follows  : 

na  :  a  :  ma;        a  :  na  :  ma;        ma  :  a  :  na; 
na  :  ma  :  a;        a  :  ma  :  na ;        ma  :  na  :  a. 

Inasmuch   as  the   signs   of  these    six    symbols   are 


THE  ISOMETRIC  SYSTEM.  49 

throughout  the  same,  all  the  planes  which  they  rep- 
resent must  belong  to  a  single  octant  (p.  31).  The 
three  principal  planes  of  symmetry  of  course  require 
the  repetition  of  the  same  group  of  six  planes  in  each 
of  the  eight  octants  into  which  they  divide  space ; 
and  hence  the  most  general  isometric  form  must  be 
bounded  by  forty-eight  planes. 

The  same  conclusion  may  be  reached  by  writing  the 
general  expression  for  the  indices  of  a  form,  { hid  \ ,  in 
every  order  and  with  every  possible  combination  of 
signs.  In  this  way  we  obtain  the  following  forty- 
eight  symbols,  each  of  which  stands  for  one  particular 
plane  of  the  most  general  isometric  form  : 


Four  upper  octants. 

hE  hki  ill  lid 

Uk  Uk  Uk  Ilk 

kU  kll  III  lU 

klh  Eh  Eh  klh 

Ihk  Ihk  Ihk  Ihk 

Ikh  Ikh  Ikh  Ikh 


Four  lower  octants. 


hkl  hkl  hkl  hE 

hlk  hlk  Uk  III 

khl  ML  khl  lU 

klh  klh  klh  klh 

Ihk  Ihk  Ihk  Ihk 

Ikh  Ikh  Ikh  Ikh 


Each  vertical  column  here  represents  the  planes  of 
a  single  octant.  The  order  of  the  octants,  commenc- 
ing with  the  upper,  right,  front  one,  is  around  the 
upper  half  of  the  crystal  from  right  to  left,  and  then, 
in  a  similar  way,  around  the  lower  half. 

The  number  of  planes  belonging  to  the  most  general 
isometric  form  may  be  arrived  at  in  still  another  way. 
The  nine  planes  of  symmetry  (see  Fig.  48)  divide 
space  into  forty -eight  equal  triangular  sectints.  The 
presence  of  any  plane  in  one  of  these  secants,  oblique 
to  all  the  planes  of  symmetry,  necessitates  another 
plane,  similarly  inclined,  in  each  of  the  other  forty- 


50  CRYSTALLOGRAPHY. 

seven  secfots.     Hence  forty- eight  is  the  number  of 
planes  bounding  the  most  general  form. 

The  parameters,  ra  and  n,  in  the  -most  general 
symbol  stand  for  any  rational  quantity  greater  than 
one  and  less  than  infinity.  The  reason  why  they  are 
never  made  less  than  one  is  because  in  the  isometric 
system  all  of  the  axes  are  equal  and  interchangeable ; 
hence  any  one  of  the  axial  intercepts  for  any  plane 
may  be  assumed  as  unity,  and  it  is  customary  to 
make  the  shortest  of  them  unity.  Now  all  the  other 
types  of  holohedral  forms  possible  in  the  isometric 
system  may  be  deduced  as  special  cases  of  the  most 
general  type,  na  :  a  :  ma,  by  giving  limiting  values  to 
one  or  both  of  its  parameters.  These  limiting  values 
are  three  in  number,  viz.,  the  smallest  possible  value, 
unity ;  the  largest  possible  value,  infinity ;  and  a  value 
for  one  parameter  equal  to  that  of  the  other.  Since 
special  cases  may  be  produced  by  limiting  either  one 
or  both  of  the  variable  parameters,  we  find  that  seven 
and  only  seven  distinct  types  of  isometric  holohedrons 
are  possible,  which  fall  naturally  into  the  three  follow- 
ing classes :  ,  / 
Class  I.  Forms  with  two  variable  parameters. 

1.  m  >  n,  general  symbol  becomes  na  :  a  :  ma. 
Class  II.  Forms  with  one  variable  parameter. 

2.  ra    or  7i  —  oo ,  general    symbol   becomes 

GO  a  :  a  :  ma. 

3.  m  or  n  =  1,  general  symbol  becomes  a  :  a  :  ma. 

4.  m  =  n,  general  symbol  becomes  ma  :  a  :  ma. 
Class  III.  Forms  with  no  variable  parameter. 

5.  m  =  1,  n  =  oo ,  general   symbol  becomes 

oo  a  :  a  :  a. 


THE  ISOMETRIC  SYSTEM. 


51 


6.  m  and  n  =  oo ,  general  symbol  becomes 

oo  a  :  a  :  oo  a. 
\    7.  m  and  n  =  1,  general  symbol  becomes  a  :  a  :  a. 

If  we  employ  the  indices  where  h>7c>l,  the  sym- 
bols corresponding  to  these  seven  types  become 
{hkl\,  {MO},  \W},  {hick},  {110},  1100},  and  {111}. 

The  particular  characters  of  these  seven  isometric 
form-types  we  shall  now  proceed  to  consider  in 
succession. 

The  Hexoctahedron.  We  shall  best  be  able  to  appre- 
ciate the  nature  of  the  most  general  or  forty-eight- 


FiQ.  50. 


sided  form  if  we  construct  upon  three  equal  axes  the 
planes  which  occupy  a  single  octant.  For  this  pur- 
pose we  may  assume  the  definite  values  n  =  2  and 


52  CRYSTALLOGRAPHY. 

m  =  3.     Then  the  possible  permutations  of  the  sym- 
bol become : 

2a  :    a  :  3a ;          a  :  2a  :  3a ;          3a  :  a  :  2a ; 
2a  :  3a  :    a ;          a  :  3a  :  2a ;          3a  :  2a  :  a . 

Each  expression  represents  one  plane  of  the  same 
octant,  which  may  be  independently  constructed  on 
the  axes.  The  intersections  of  the  six  planes  thus 
obtained  give  the  assemblage  repre- 
sented in  Fig.  50.  The  principal 
planes  of  symmetry  now  require 
the  occurrence  of  a  similar  group 
in  each  of  the  other  seven  octants, 
and  the  result  is  the  solid  figure 
shown  in  Fig.  51.  Each  plane  of  the 
fundamental  form  is  here  replaced 
by  a  hexagonal  pyramid,  whence  the  name  for  this 
form  is  the  hexoctahedron. 

This  form  is  bounded  by  forty-eight  similar  scalene 
triangles.  Its  solid  angles  are  of  three  kinds:  six 
octahedral,  at  the  extremities  of  the  principal  axes ; 
eight  hexahedral,  at  the  extremities  of  the  trigonal 
axes ;  and  twelve  tetrahedral,  at  the  extremities  of  the 
digonal  axes.  There  are  also  three  kinds  of  combina- 
tion-edges, twenty-four  of  each.  Of  these  the  shorter, 
for  reasons  to  be  explained  beyond,  may  be  called  the 
cubic  edges  (c,  Fig.  51) ;  those  of  intermediate  length, 
the  octahedral  edges  (o,  Fig.  51) ;  and  the  longer,  the 
dodecahedral  edges  (d,  Fig.  51).  Naumann's  symbol  for 
this  form  is  m  0  n ;  Dana's,  m-n. 

The  Tetrahexahedron.  If  one  of  the  two  variable 
parameters  be  given  the  limiting  value  infinity,  the 
general  formula  becomes  oo  a  :  a  :  ma.  This  repre- 


THE  ISOMETRIC  SYSTEM. 


53 


sents  a  figure  bounded  by  twenty-four  sides,  each  of 
which  is  parallel  to  one  of  the  principal  axes.  It  is 
therefore  a  combination  of  planes  of  the  prismatic 
type  (p.  36).  We  may  make  a  construction  of  the 
planes  in  one  octant,  as  was  done  in  the  last  case,  as- 
suming for  m  the  value  3.  The  possible  permutations 
then  become : 


oo  a  :    a  :  3a  ; 

QO  a  :  3a  :    a ; 


a  :  co  a  :    3a ; 
a  :    3a  :  co  a ; 


3a  :  GO  a  :      a ; 

3a  :      a  :  oo  a . 


FIG.  52. 


The  resulting  group  of  planes  is  not,  at  first  glance,  very 
different  from  that  obtained  before ;  but  here  each  plane 
is  normal  to  a  principal  plane  of  symmetry,  and  there- 


54 


CRYSTALLOGRAPHY. 


fore  belongs  equally  to  two  contiguous  octants  (Fig. 
52).  The  complete  form  (Fig.  53)  is 
like  a  cube  whose  faces  are  replaced 
by  a  quadratic  pyramid,  and  it  is 
therefore  called  the  tetrahexahedron. 

The  planes  of  this  form  are  twenty- 
four  similar  isosceles  triangles.      Its 
Fio753.  solid  angles  are  of  two  kinds  :    six 

tetrahedral,  at  the  ends  of  the  principal  axes;  and 
eight  hexahedral,  at  the  ends  of  the  trigonal  axes. 
Twelve  cubic  and  twenty-four  dodecahedral  edges  re- 
main, but  the  octahedral  edges  have  disappeared  by 
becoming  angles  of  180°.  Naumann's  symbol  for  this 
form  is  oo  0  n  ;  Dana's  is  i-n. 

The  Trisoctahedron.  Giving  n  its  smaller  instead  of  its 
larger  limit  changes  the  general  formula  to  a  :  a  :  ma. 
Of  this  only  three  permutations  are  possible,  which 
indicates  that  the  resulting  form  has 
but  three  planes  in  an  octant.  A 
construction  similar  to  those  given 
in  the  preceding  cases  can  be  readily 
made  by  the  student.  This  pro- 
duces a  group  of  planes,  which,  when 
developed  for  each  octant,  results  FIG.  54. 

in  the  form  shown  in  Fig.  54.  It  resembles  an  octa- 
hedron with  each  of  its  planes  replaced  by  a  triangu- 
lar pyramid,  whence  its  name — the  trisoctahedron. 

The  planes  of  this  form  are  twenty-four  similar 
isosceles  triangles,  each  normal  to  a  plane  of  symme- 
try, and  hence  monosymmetric  (p.  43).  Its  solid  angles 
are  of  two  kinds :  six  octahedral,  at  the  ends  of  the 
principal  axes  ;  and  eight  trihedral,  at  the  ends  of  the 
trigonal  axes.  There  are  twelve  octahedral  and  twenty- 


THE  ISOMETRIC  SYSTEM.  55 

four  dodecahedral  edges,  while  the  cubic  edges  have 
disappeared,  like  the  octahedral  edges  in  the  tetra- 
hexahedron.  Naumann's  symbol  for  this  form  is  mO  ; 
Dana  designates  it  by  m,  and  calls  it  the  trigonal  tris- 
octahedron. 

The  Icositetrahedron.  When  n  =  m>  the  general 
formula  becomes  ma  :  a  :  ma,  of 
which  there  are  again  but  three 
permutations  possible.  A  construc- 
tion like  the  others,  with  some 
definite  value  assumed  for  ra,  yields 
three  planes  in  each  octant,  and  a 
completed  figure  like  that  shown  in 
Fig.  55.  In  allusion  to  its  being  twenty-four-sided, 
it  is  usually  called  the  icositetrahedron* 

The  planes  of  this  form  are  similar  trapeziums.  Its 
solid  angles  are  of  three  kinds :  six  tetrahedral  at  the 
ends  of  the  principal  axes ;  twelve  tetrahedral  at  the 
ends  of  the  digonal  axes ;  and  eight  trihedral  at  the 
ends  of  the  trigonal  axes.  There  are  twenty-four 
octahedral  and  twenty-four  cubic  edges,  while  the 
dodecahedral  edges  have  disappeared.  Naumann's 
symbol  for  this  form  is  raOm;  Dana's  is  m-m. 

The  Rhombic  Dodecahedron.  If  one  parameter  is  given 
its  largest,  and  the  other  its  smallest  limiting  value, 
the  general  formula  becomes,  a  :  a  :  oo  a.  This  is 
capable  of  three  permutations,  which  locates  three 
planes  in  each  octant ;  but  the  sign  of  infinity  shows 
that  each  plane  is  parallel  to  an  axis,  and  therefore 
common  to  two  contiguous  octants.  The  result  must 

*  It  is  also  known  as  the  trapezohedron,  the  leucitohedron,  and 
tetragonal  trisoctahedron, 


56 


CRYSTALLOGRAPHY. 


FIG.  56. 


be  a  form  bounded  by  twelve  planes,  which  the  con- 
struction shows  are  similar  rhombs 
(Fig.  56).  This  form  is  called  the 
rhombic  dodecahedron.  It  has  six 
acute  tetrahedral  angles  at  the  ex- 
tremities of  the  principal  axes,  and 
eight  obtuse  trihedral  angles  at  the 
ends  of  the  trigonal  axes.  Its  edges 
are  twenty-four  in  number,  and  are 
all  of  one  sort  (dodecahedral),  enclosing  angles  of  120°. 
Each  of  its  faces  is  normal  to  two  planes  of  symmetry, 
and  therefore  disymmetric.  Naumann's  symbol  for 
this  form  is  GO  0 ;  Dana's  is£ 

The  Hexahedron  (Cube).  If  both  parameters  reach 
their  maximum  limit,  the  general  formula  becomes 

oo  a  :  a  :  oo  a,  also  capable  of  three          ^ 

permutations.       Hence  there  are 

three   planes  in  each  octant,  but 

the    two    signs   of    infinity  show 

that  every  plane  is  parallel  to  two 

axes,  and  hence  is  common  to  four 

contiguous    octants.     The    result 

is   a   six-sided    figure,    which    is 

therefore  called  the   hexahedron  (Fig.  57).      Its   sides 

are    squares,   normal    to   four  planes   of   symmetry, 

and  quadrisymmetric.    It  has  eight  similar  trihedral 

angles  at  the  ends  of  the  trigonal   axes,  and   twelve 

similar   edges  (cubic)  enclosing  angles  of  90°.     Nau- 

mann's  symbol  for  the  cube  is  oo  0  oo;  Dana's  is  i-i. 

The  Octahedron.  The  simplest  form  to  which  the 
most  general  isometric  parameter  symbol  can  be  re- 
duced is  a  :  a  :  a,  which  represents  one  plane  in  each 
of  the  eight  octants,  intersecting  all  of  the  axes  at  unit 


FIG.  57. 


THE  ISOMETRIC  SYSTEM.  57 

distance  from  the  centre.  This  produces  the  regular 
octahedron,  or  fundamental  form  of  the  system  (see  p. 
47,  Fig.  49).  Its  faces  are  all  equilateral  triangles  ;  its 
six  solid  angles  all  similar  and  tetrahedral,  situated  at 
the  extremities  of  the  axes  ;  and  its  edges  are  all  simi- 
lar (octahedral).  Naumann  abbreviates  the  parameter 
symbol  of  this  form  into  its  initial  letter,  0;  while 
Dana  designates  it  by  the  figure  1. 

Relations  of  the  Seven  Isometric  Holohedrons  to  each 
other — Limiting  Forms.  We  have  seen  (p.  50)  that  the 
six  simpler  types  of  isometric  holohedrons  may  be  re- 
garded as  special  cases  of  the  most  general  type  or 
hexoctahediion.  Of  course  those  types  whose  symbols 
contain  one  or  more  variables  may  be  represented  by 
a  variety  of  forms,  which  are  alike  in  the  number, 
distribution,  and  symmetry  of  their  faces,  but  which 
differ  in  their  corresponding  interfacial  angles.  Thus 
202,  303,  and  505  all  represent  icositetrahedrons, 
although  their  corresponding  angles  are  not  iden- 
tical. On  the  other  hand,  only  one  representative  is 
possible  of  those  types  whose  symbols  contain  no 
variable  parameter ;  and  hence  all  cubes,  octahedrons, 
and  rhombic  dodecahedrons  must  be  alike.  These 
are  therefore  called  fixed  forms. 

In  proportion  as  the  parameters  in  any  symbol 
approach  their  limiting  values,  so  do  the  forms  which 
they  represent  approach  nearer  to  their  limiting  forms. 
Plate  I.  is  arranged  to  show  the  relations  of  the  iso- 
metric holohedrons  in  this  respect.  The  three  fixed 
or  un variable  forms  occupy  the  corners  of  the  tri- 
angular diagram,  since  they  are  the  final  limits  between 
which  the  other  types  vary.  The  colors  of  their  edges 
— octahedral  (black),  cubic  (blue),  and  dodecahedral 


58  CRYSTALLOGRAPHY. 

(red) — are  retained  in  all  the  other  forms.  Between 
each  pair  of  fixed  forms  oscillates  one  of  the  three 
types  of  twenty-four-sided  figures  which  have  but  a 
single  variable  parameter  in  their  symbols. 

Let  us,  as  an  illustration,  consider  the  case  of  the 
trisoctahedron.  The  number  of  trisoctahedrons  theo- 
retically possible  is  infinite,  because  ra  may  be  given 
an  infinite  number  of  rational  values ;  practically  the 
number  is  small  because,  on  natural  crystals,  m  is 
generally  some  small  whole  nu"mber.  Such  a  sequence 
of  trisoctahedrons  may  be  represented  by  the  following 
symbols :  £0,  f  0,  f  0,  20,  f  0,  30,  40,  60,  120,  180, 
etc.  As  the  value  of  m  decreases,  the  more  obtuse  do 
the  dodecahedral  (red)  edges  become ;  until  when 
m  reaches  its  lower  limit,  1,  the  angles  disappear  by 
becoming  =  180°,  and  the  form  merges  into  the  octa- 
hedron— one  of  its  limiting  forms.  As  the  value  of 
m  increases,  the  more  acute  the  dodecahedral  edges 
of  the  trisoctahedron  and  the  more  obtuse  its  octa- 
hedral edges  become.  When  m  reaches  its  upper  limit, 
oo ,  the  former  have  become  120°  and  the  latter  180°, 
i.e.,  the  trisoctahedron  has  graded  into  the  rhombic 
dodecahedron,  the  other  of  its  limiting  forms. 

Exactly  the  same  gradations  may  be  traced  for  the 
sequence  of  icositetrahedrons  between  its  limiting 
forms,  the  octahedron  and  the  cube ;  and  for  the  pos- 
sible tetrahexahedrons,  between  the  cube  and  dodeca- 
hedron. 

Each  type  of  the  twenty-four-sided  figures  possesses 
the  two  sorts  of  edges  belonging  to  its  own  limiting 
forms.  Each  can  oscillate  backward  or  forward  along 
a  single  direction,  because  it  has  but  a  single  variable 
in  its  symbol. 


THE  ISOMETRIC  SYSTEM.  59 

The  variations  of  the  hexoctahedral  type,  with  two 
variable  quantities  in  its  symbol,  na  :  a  :  ma,  are  more 
complex.  The  value  of  one  parameter  may  be  changed, 
while  the  other  remains  fixed ;  or  both  may  be  changed 
simultaneously.  Thus  sequences  of  hexoctahedrons 
may  be  deduced,  oscillating  along  different  lines,  and 
between  different  pairs  of  limiting  forms. 

Suppose  we  start  with  the  hexoctahedron  402,  we 
may  vary  but  one  parameter  in  three  ways :  (1)  By 
increasing  the  greater:  402,  502,  602,  1202,  1802, 
etc.,  to  QO  02,  the  tetrahexahedron,  one  limiting  form. 
(2)  By  decreasing  the  greater  to  the  limit  of  the  less  : 
4  02, 1 02,  302,  f  02,  etc.,  to  202,  the  icositetrahedron, 
another  limiting  form.  (3)  By  decreasing  the  less,  402, 
40J,  4  Of,  4  Of,  etc.,  to  40,  the  trisoctahedron,  a  third 
limiting  form. 

Both  parameters  may  also  be  varied  simultaneously 
in  three  ways  :  (1)  Both  maybe  increased  :  402,  503, 
604,  806,  etc.,  to  oo  0  oo,  the  cube.  (2)  Both  may  be 
diminished:  402,  |OJ,  3 Of,  fOf,  to  0,  the  octahe- 
dron. (3)  One  may  be  increased,  while  the  other  is 
diminished  :  402,  50J,  6  Of,  7  Of  to  oo  0,  the  rhombic 
dodecahedron. 

Thus  we  see  that  all  the  other  holohedral  forms  of 
the  isometric  system  are  not  merely  special  cases  of 
the  hexoctahedron,  but  that  they  are  as  well  its  limit- 
ing forms  in  different  directions. 

The  Holohedral  Isometric  Forms  in  Combination.  Al- 
though all  isometric  forms  completely  enclose  space, 
and  are  therefore  capable  of  independent  occurrence, 
still  most  isometric  crystals  exhibit  the  occurrence  of 
two  or  more  forms  together.  These  combinations  are 
far  too  manifold  for  special  description.  Their  ac- 


60 


CR  YSTALLOGRAPHY. 


quaintance  must  be  made  by  practice  with  models  and 
natural  crystals. 

The  figures  on  Plate  II.  are  arranged  to  show  each 
of  the  isometric  holohedrons  in  two  different  com- 
binations with  all  the  others.  The  simple  forms  run 
in  a  diagonal  through  the  plate.  The  combinations 
above  this  line  show  the  simpler,  and  those  below  it 
the  more  complex  forms  predominating.* 

Only  a  few  points  relating  to  these  combinations  can 
be  specified  here.  We  notice  that  the  cube  and  octa- 
hedron mutually  truncate  each  other's  solid  angles, 
while  the  edges  of  both  forms  are  replaced  by  the 


FIG.  56. 


FIG.  59. 


planes  of  the  rhombic  dodecahedron.  In  combinations 
of  fixed  with  variable  forms,  or  of  variable  forms  with 
each  other,  the  particular  mode  of  replacement  depends 
on  the  values  of  the  parameters.  For  instance,  the 
faces  of  the  icositetrahedron  truncate  the  edges  of  the 
rhombic  dodecahedron  only  when  the  parameters  m, 
of  the  former,  are  equal  to  two  (PL  II.  fig.  18).  If  these 
parameters  are  greater  than  two,  its  faces  replace  the 
tetrahedral ;  if  less  than  two,  the  trihedral  angles  of  the 
dodecahedron  (Figs.  58  and  59).  Similar  relations, 


*  From  Ulrich's  Krystallographische  Figurtaf eln.  4° ;  Hannover, 

1884. 


THE  ISOMETRIC  SYSTEM. 


61 


which  will  be  readily  understood,  are  indicated  by 
dotted  lines  on  Figs.  18,  26,  27,  39  and  46,  of  Plate  II. 
Combinations  of  three  or  more  forms  are  quite  as 
common  as  those  of  two.  A  few  examples  of  such 
complex  combinations  are  given  in  the  following  fig- 
ures : 


FIG.  60. 


FIG.  61. 


Fig.  60  (galena)  shows  the  cube  (h),  the  octahedron 
(o),  and  the  dodecahedron  (d). 

Fig.  61  (garnet)  shows  the  dodecahedron,  the 
icositetrahedron,  202,  J211}  (Z),  the  tetrahexahedron, 
oo  02,  J210J  (e),and  the  hexoctahedron,  3  Of ,  J231J  (s). 

Fig.  62  (amalgam)  shows  the  cube, 
the  dodecahedron,  the  tetrahexahe- 
dron, oo 03,  {310|  (/),and  the  icosi- 
tetrahedron, 202,  1 211}  (Z). 

The  following  isometric  sub- 
stances may  be  mentioned  as  exam- 
ples of  holohedral  crystallization : 
the  metals  copper,  silver,  gold,  and 
platinum ;  amalgam  (HgAg) ;  the  sulphides  of  lead 
(galena)  and  of  silver  (argentite) ;  the  chlorides  of 
sodium  (halite)  and  of  silver  (cerargyrite) ;  calcium 
fluoride  (fluor-spar) ;  spinel,  garnet,  microlite,  sodalite, 
nosean,  leucite,  and  analcite. 


FIG.  62. 


62  CRYSTALLOGRAPHY. 

HEMIHEDRAL  DIVISION  OF  THE  ISOMETRIC  SYSTEM. 

Kinds  of  Hemihedrism.  As  has  been  explained  in  the 
preceding  chapter,  one  half  of  the  planes  composing  a 
complete  form  may  occur  on  crystals  of  certain  sub- 
stances independently  of  the  other  half.  Such  partial 
forms  are,  however,  only  possible  when  the  planes 
which  compose  them  are  selected  from  the  corre- 
sponding complete  form  in  accordance  with  certain 
fixed  conditions,  viz.:  they  must  intersect  the  extremi- 
ties of  all  equivalent  axes  of  symmetry  in  the  same 
number,  under  equal  angles  and  at  equal  distances 
from  the  centre  (p.  41). 

In  order  to  discover  how  many  different  kinds  of 
hemihedral  forms  are  possible  in  the  isometric  system, 
we  may  imagine  one  half  of  the  planes  bounding  the 
most  general  form — the  hexoctahedron — to  be  selected 
in  every  conceivable  way,  and  then  note  which  of  these 
ways  satisfy  the  above  conditions.  Such  a  method  of 
procedure  shows  that,  although  we  can  choose  one 


FIG.  63.  FIG.  64.  FIG.  65. 

half  of  the  forty-eight  planes  of  the  most  general 
holohedron  in  a  great  number  of  different  ways,  only 
three  of  them  yield  crystallographically  possible  partial 
forms.  These  three  methods  of  selection,  each  capable 
of  producing  a  different  kind  of  isometric  hemihedrism, 
are  illustrated  in  the  three  figures,  63-65.  The  first 


THE  ISOMETRIC  SYSTEM.  63 

(Fig.  63)  is  a  selection  by  alternate  planes  ;  the  second 
(Fig.  64)  is  by  pairs  of  planes  which  intersect  in  the 
principal  planes  of  symmetry,  or  in  the  octahedral 
edges ;  and  the  third  (Fig.  65)  is  a  selection  by  alter- 
nate octants.  If  we  imagine  either  the  white  or  shaded 
planes  of  these  figures  to  disappear,  and  the  others  to 
be  extended  until  they  intersect,  three  new  forms  will 
result,  all  of  which  satisfy  the  conditions  of  hemihe- 
drism,  and  which  occur  on  natural  crystals.  Thus  three 
distinct  kinds  of  isometric  hemihedrism  are  possible. 
These  are  called,  for  reasons  which  will  appear  as  each 
in  turn  is  considered : 

1.  Gyroidal,  or  plagiohedral  hemihedrism  ; 

2.  Pentagonal,  or  parallel-face  hemihedrism  ; 

3.  Tetrahedral,  or  inclined-face  hemihedrism. 
Gyroidal  Hemihedrism.     If  the  alternate  planes  of  the 

most  general  isometric  form — the  hexoctahedron — are 


FIG.  66.  FIG.  67. 

extended  until  they  intersect,  a  new  form  will  result 
bounded  by  twenty-four  similar  but  unsymmetrical 
pentagons.  Two  such  forms  must  be  derivable  from 
every  hexoctahedron,  one  produced  from  one  set  and 
the  other  from  the  other  set  of  alternate  planes  (Figs. 
66  and  67).  They  are  called  pentagonal  icositetrahedrons, 
and  are  distinguished  as  right-  and  left-handed  ac- 


64  CRYSTALLOGRAPHY. 

cording  as  they  contain  the  right  or  left  top  plane  of 
the  front,  upper  octant.  Their  symbols,  in  the  nota- 
tions of  Nauinann  and  Miller,  are 

mOn  .W\  mOn, 

and    -     l> 


where  h  >  k  >  ?,  as  in  the  case  of  holohedral  forms 
(p.  51). 

Apparently  Holohedral  Hemihedrons.  Since  all  the 
other  six  isometric  holohedrons  may  be  regarded  as 
special  cases  of  the  hexoctahedron  (p.  50),  we  may 
consider  them  all  as  forty  -eight-sided  figures,  certain 
of  whose  planes  intersect  at  angles  of  180°.  From  this 
point  of  view,  the  faces  of  the  cube  are  composed  of 
eight  planes  ;  those  of  the  octahedron  of  six  ;  those  of 
the  dodecahedron  of  four  ;  etc.  Now  if  the  above 
method  of  selection  be  applied  to  the  forty-eight 
planes,  by  which  every  isometric  holohedron  may  be 
considered  as  bounded,  it  is  evident  that  in  every 
case,  except  the  most  general,  the  surviving  planes 
will,  by  their  extension,  reproduce  the  form  without 
geometrical  change.  This  must  be  so,  because  every 
face  of  the  six  more  special  forms  is  m'ade  up  of  at 
least  two  contiguous  planes  of  the  general  form,  in- 
tersecting at  angles  of  180°  ;  and  hence  either  half  of 
any  face  is  enough  to  reproduce  it.  This  may  be  seen 
from  the  six  following  figures,  whose  planes  are  shaded 
in  accordance  with  the  alternate  method  of  selection. 
Thus  only  one  geometrically  new  form-type  is  possi- 
ble by  gyroidal  hemihedrism,  but  other  apparently 
holohedral  forms  combined  with  it  must  be  regarded 
as  just  as  truly  hemihedral.  Their  essential  character 
consists  in  their  molecular  structure,  and  this  must 


THE  ISOMETRIC  SYSTEM. 


65 


be  the  same  throughout  the  same  crystal  individual. 
If  a  given  molecular  structure  can  produce  any  form 
which  is  geometrically  hemihedral,  this  is  a  proof  that 
all  the  forms  produced  by  the  same  structure  must  be 
equally  hemihedral,  whether  they  have  a  different 


FIG.  71. 


FIG.  72. 


FIG.  73. 


shape  from  holohedral  forms  or  not.  Thus  we  may 
have  two  cubes  or  octahedrons  which  are  outwardly 
alike  but  in  reality  different ;  and  their  difference  can, 
in  many  cases,  be  demonstrated  by  &ieir  physical  be- 
havior or  by  etching. 

Symmetry  of  Gyroidal  Forms.  An  inspection  of  the 
only  new  gyroidal  hemihedron — the  pentagonal  icosi- 
tetrahedron — shows  that  it  possesses  no  plane  of  sym- 
metry whatever.  All  of  the  nine  isometric  planes  of 
symmetry  disappear  by  the  alternate  method  of  selec- 
tion. Nevertheless,  the  faces  of  this  form  are  refer- 
able to  three  equal  and  rectangular  axes,  and  it  is 


66 


CRYSTALLOGRAPHY. 


therefore  to  be  reckoned  in  the  isometric  system. 
Although  the  right-  and  left-handed  pentagonal  icosi- 
tetrahedrons  themselves  are  without  symmetry,  they 
are  still  symmetrical  with  reference  to  each  other. 
Symmetrical  forms,  which  are  themselves  devoid  of 
planes  of  symmetry  and  cannot  therefore  be  brought 
by  any  revolution  into  exactly  the  same  position, 
are  called  in  crystallography  enantiomorphous  (from 
evavTioS,  opposite,  and  ^6pcf)rj^  form). 

In  spite  of  their  apparently  holohedral  shape,  all 
the  other  gyroidal  forms  of  the  isometric  system 
must  also  be  considered  as  devoid  of  symmetry,  be- 
cause they  are  produced  by  an  asymmetric  molecu- 
lar arrangement. 

Gyroidal  hemihedrism  is  not  of  common  occur- 
rence, as  it  has,  up  to  the  present  time,  been  observed 
on  the  crystals  of  only  three  substances :  cuprite 
(Cu2O),  sylvine  (KC1),  and  sal-ammoniac  (NH4C1). 

Pentagonal  Hemihedrism.  The  second  kind  of  possi- 
ble isometric  hemihedrism  is,  as  we  have  seen  (p.  63), 


FIG 


FIG.  75. 


produced  by  the  selection  of  one  half  of  the  planes  of 
the  hexoctahedron  by  pairs  which  intersect  in  the 
principal  planes  of  symmetry,  or  octahedral  edges. 

The  survival  and  extension  of  the  planes  belonging 
to  these  two  sets  of  alternating  pairs  produce  from 


THE  ISOMETRIC  SYSTEM. 


67 


the  most  general  form  two  new  figures,  each  bounded 
by  twenty-four  similar  trapeziums  (Figs.  74  and  75). 
These  forms  possess  three  sorts  of  solid  angles  and 
three  sorts  of  edges.  The  two  figures  produced  by 
the  survival  of  the  two  halves  of  the  planes  of  any 
hexoctahedron  differ  from  each  other  only  in  their 
position.  Either  one  may,  by  a  revolution  of  90°  about 
any  of  its  axes,  be  brought  exactly  into  the  posi- 
tion of  the  other.  Such  figures,  in  contradistinction 
to  enantiomorphous  forms,  are  said  to  be  congruent. 
Their  positions  are  distinguished  as  positive  and  nega- 
tive. The  above-described  hemihedrons  are  called  di- 
dodecahedrons  or  diploids.  Their  symbols  according 
to  Naumann  and  Miller  are 


n 

J,   7t 


and 


[~mOn~] 
~~2~J'     n 


Any  other  isometric  form  is  capable  of  producing  a 
geometrically  new  hemihedron  by  this  method  of  selec- 


FIG.  76. 


FIG.  77. 


tion,  if  its  planes  correspond  exactly  to  pairs  of  planes 
selected  on  the  hexoctahedron.  This  is  the  case  with 
the  faces  of  the  tetrahexahedron,  but  with  those  of 
no  other  isometric  form.  The  selection  of  alternate 


68  CRYSTALLOGRAPHY. 

planes  of  the  tetrahexahedron  produces  two  new  and 
congruent  forms,  bounded  by  twelve  similar  but  un- 
equilateral  pentagons  (Figs.  76  and  77).  These  are 
known  as  the  positive  and  negative  pentagonal  do- 
decahedrons, or,  on  account  of  their  frequent  occur- 
rence on  crystals  of  pyrite  (FeS2),  as  pyritohedrons. 
Their  symbols  are  written  by  Naumann  and  Miller 


and       - 


Apparently  Holohedral  Hemihedrons.  On  no  other  iso- 
metric holohedron,  except  the  two  just  mentioned,  do 
the  planes  correspond  exactly  to  the  pairs  of  faces 
selected  on  the  most  general  form  ;  therefore,  which- 


FIG.  7d.  FIG.  79.  FIG.  80. 

ever  pair  is  selected,  portions  of  all  the  faces  will 
survive  on  the  five  other  holohedrons,  and  these,  by 
their  extension,  will  reproduce  the  forms  as  they  were 
before.  This  is  shown  by  the  five  figures,  78-82, 
which  are  shaded  to  correspond  to  the  pentagonal 

*  The  regular  dodecahedron  of  geometry,  with  its  angles  all  equal 
and  its  faces  equilateral  pentagons,  is  not  crystallographically  possi- 
ble, because  its  indices  would  be  2  . 1  -|-  4/0 .  0,  involving  an  irrational 
quantity.  The  form  whose  indices  are  7TJ580J  approaches  it  very 
closely. 


THE  ISOMETRIC  SYSTEM. 


selection.     As  explained  in  the  case  of  gyroidal  hemi- 
hedrism,  forms  of  this  character  are  to  be  regarded 


FIG.  81. 


FIG.  82. 


as  truly  hemihedral  as  though  they  produced  com- 
pletely new  shapes. 

Symmetry  of  Pentagonal  Forms.  An  examination  of 
the  two  new  form-types  produced  by  this  kind  of 
hemihedrism  shows  that  they  possess  three  planes  of 
symmetry,  corresponding  in  their  position  to  the  three 
principal  planes  of  symmetry  of  holohedral  forms. 
These  are,  however,  no  longer  principal  planes  of 
symmetry,  because  they  do  .not  contain  strictly  inter- 
changeable directions  (p.  34).  The  six  secondary 
planes  of  symmetry  belonging  to  holohedral  forms 
have  disappeared. 

The  pentagonal  hemihedrism  is  also  called  paraUd- 
face,  because  the  planes  of  its  forms,  like  those  of  holo- 
hedrons,  are  arranged  in  parallel  pairs.  With  the 
succeeding  kind  of  hemihedrism  this  is,  however,  not 
the  case,  so  that  this  is  called  inclined-face,  as  has  been 
already  explained  on  p.  42  of  the  preceding  chapter. 

Pentagonal  Hemihedral  Forms  in  Combination.  On  ac- 
count of  the  common  occurrence  of  parallel-face  hemi- 
hedrism in  the  isometric  system,  it  will  be  well  to 


7. 


(Fig.    85>     This   solid 

exaetl  j  represented  bj  anj  crystal, 

wouM  hare  irrational  indices.    Fig.  86  sbows  a 


iteelf   be 


combination  to  the  last,  where  the  octahedron  is  re- 
placed bj  the  diploid.    Fig.  87  is  an  example  of  a 


THE  ISOMETRIC  SYSTEM.  71 

cube  whose  solid  angles  are  replaced  by  the  trihedral 
angles  of  the  diploid,  [-jpj  ,  WJ231}. 

As  examples  of  parallel-face  hernihedrism  may  be 
mentioned  stannic  iodide,  SnI4 ;  iron  disulphide,  FeSa 
(pyrites) ;  cobalt  arsenide,  CoAs,  (cobaltite) ;  cobalt- 
glance,  (Co,Fe)AsS;  alum,  RR(SO4)2  + 12  aq. 

Inclined-face  or  Tetrahedral  Hemihedrism.  The  selec- 
tion of  planes  by  alternate  octants  will  produce  geo- 
metrically new  forms  from  all  holohedrons  whose  faces 
belong  exclusively  to  single  octants.  A  glance  at 
Plate  I  will  show  that  there  are  four  types  of  isometric 
forms  of  which  this  is  true  :  the  hexoctahedron,  trisoc- 
tahedron,  icositetrahedron,  and  octahedron. 

The  hexoctahedron  yields,  in  this  way,  two  congru-s 
ent  half -forms,  bounded  by  twenty-four  similar  scalene 
triangles  (Figs.  88  and  89).  Both  edges  and  solid  angles 


FIG.  88.  Fro.  89. 

are  of  three  kinds.     These  forms  are  called  hextetra- 
hedrons,  and  their  symbols  are  written 

mOn  mOn 

and         --  "- 


The  trisoctahedron,  by  the  same  method  of  selection 
of  its  planes,  yields  two  congruent  half-forms,  bounded 


72 


CRYSTALLOGRAPHY. 


by  twelve  similar  trapeziums,   intersecting  in   three 
kinds  of  solid  angles  and  two  sorts  of  edges  (Figs.  90 


FIG.  90. 


FIG.  91. 


and  91).     They  are  known  as  tetragonal  tristetrcihedrons, 
their  symbols  being 


and        - 


The  corresponding  half-forms  developed  from  the 
icositetrahedron  are  bounded  by  twelve  similar  isos- 
celes triangles,  intersecting  in  two  kinds  of  edges  and 


FIG.  92.  FIG.  93. 

solid  angles  (Figs.  92  and  93).     These  are  called  trigo- 
nal tristetrahedrons*  their  symbols  being 

mOm  mOm  - 

,    K\hkk\. 


and 


*  The  trigonal  tristetrahedron  is  also  called  the  trigondodecahedron 
or  pyramid-tetrahedron;  while  the  tetragonal  tristetrahedron  is  known 
as  the  deltoid  dodecahedron.  It  is  well  to  call  attention  to  the  resem- 


THE  ISOMETRIC  SYSTEM.  73 

The  octahedron  yields  two  congruent  half-forms, 
which  are  the  regular  tetrahedrons  of  geometry.  They 
are  bounded  by  four  equilateral  triangles,  intersecting 
in  six  similar  edges  and  four  equal  trihedral  angles 


FIG.  94.  FIG.  95. 

(Figs.  94  and  95).  These  are  the  simplest  crystal 
forms  in  any  system  which  completely  enclose  space. 
Their  symbols  are 

+^,    /cjlllj       and          -\,     K\\l\\. 

Apparently  Holohedral  Hemihedrons.  It  is  evident 
that  the  three  other  isometric  holohedrons  (tetrahexa- 
hedron,  rhombic  dodecahedron  and  cube)  whose  planes 
belong  equally  to  two  contiguous  octants  can  produce 
no  new  forms  in  the  tetrahedral  hemihedrism,  because 
the  parts  of  their  planes  which  disappear  in  one  octant 
are  reproduced  by  the  extension  of  the  portions  that 

blance  between  the  planes  of  the  latter  hemihedron  and  those  of  the 
holohedron  from  which  the  former  is  derived  (icositetrahedron);  and 
vice  versa.  If  this  cross-resemblance  is  overlooked,  confusion  is  apt 
to  arise  in  remembering  the  forms  from  whjcji  these  two  hemihedrons 
are  developed, 


74 


CRYSTALLOGRAPHY. 


remain  in  the  next  octant,  and  so  on.  This  will  become 
clear  by  an  inspection  of  the  following  figures  (96,  97 
and  98). 


FIG.  96. 


FIG.  97. 


FIG.  98. 


Symmetry  of  Tetrahedral  Forms.  In  the  geometrically 
new  forms  which  are  produced  by  the  inclined-face 
or  tetrahedral  hemihedrism  the  six  secondary  planes 
of  symmetry  of  the  isometric  system  remain,  while  its 
three  principal  planes  of  symmetry  disappear.  This 
may  be  most  easily  seen  in  the  case  of  the  tetrahe- 
dron (Figs.  94  and  95),  all  of  whose  six  interfacial 
angles  are  bisected  by  secondary  planes  of  symmetry, 
corresponding  in  their  positions  to  the  faces  of  the 
rhombic  dodecahedron. 

.     Tetrahedral  Forms  in    Combination.      A  few  of  the 
more  frequent  tetrahedral  combinations  may  be  men- 


FIG.  99. 


FIG.  100. 


tioned.     Fig.  99  shows  the  cube  modified  by  the  tetra- 
hedron.  Both  forms  are,  of  course,  equally  hemihedral, 


THE  ISOMETRIC  SYSTEM. 


75 


although  the  cube  is  not  geometrically  different  from 
its  corresponding  holohedron.  Fig.  100  shows  a  posi- 
tive and  negative  tetrahedron  in  combination  ;  Fig.  101, 
a  tetrahedron  (o),  cube  (A),  and  dodecahedron  (d).  Fig. 
102  shows  a  tetrahedron  (o),  its  edges  bevelled  by  the 

202 

trigonal  tristetrahedron,  -| — ~— ,  /cj211J  (Z),  and  its  an- 
gles replaced  by  the  rhombic  dodecahedron  (d).  Fig. 


FIG.  101. 


FIG.  102. 


103   shows  the   rhombic  dodecahedron  (d)  combined 

303 
with  the  trigonal   tristetrahedron   +  -~— ,  /c{311|  (q), 

as  it  sometimes  occurs  on  zinc-blende.     Fig.  104  gives 
another  combination,  observed  on  the  mineral  boracite. 


FIG.  103. 


It  shows  the  rhombic  dodecahedron  (d),  cube  (h),  posi- 
tive and  negative  tetrahedrons  (o  and  o'),  and  negative 

202 
trigonal  tristetrahedron,    —  — — ,  K  \  211 }  (I). 


76  CRYSTALLOGRAPHY. 

As  prominent  examples  of  tetrahedral  crystalliza- 
tion may  be  mentioned :  the  diamond ;  zinc  sulphide, 
ZnS  (zinc-blende) ;  sulphide  and  selenide  of  mer- 
cury, HgS  (metacinnabarite)  and  HgSe  (tiemannite); 
tetrahedrite  R8(AsSb)2ST ;  magnesium  chloroborate, 
Mg7ClaB18O80  (boracite);  and  the  sulpho-silicate,  hel- 
vine. 

Limiting  Forms  of  Isometric  Hemihedrons.  Hemihe- 
dral  forms  which  have  a  variable  parameter  oscillate 
between  limiting  forms,  like  the  corresponding  holo- 
hedrons  (p.  58).  The  pentagonal  dodecahedron,  for 
instance,  approaches  the  cube  in  proportion  as  its  pa- 
rameter, m,  is  increased ;  and  the  rhombic  dodecahe- 
dron, in  proportion  as  it  is  diminished.  So  the  trigonal 
tristetrahedron  oscillates  between  the  tetrahedron  and 
the  cube ;  and  the  tetragonal  tristetrahedron,  between 
the  tetrahedron  and  rhombic  dodecahedron.  All  three 
hemihedral  derivatives  of  the  hexoctahedron  vary  be- 
tween the  limits  of  all  the  other  isometric  forms. 
These  relations  are  indicated  in  the  three  following 
diagrams : 

1.  Gyroidal  Hemi-  2.  Pentagonal  Hemi-  8.  Tetrahedral  Hemi- 

hedrism.  hedrism.  hedrism. 


\ 
'T*^'-    \  /     xtv-^    \  /     &r°" 

f    \\  /-"r^-i  \\  /^ 

o — 


TETAETOHEDBAL  DIVISION  OF  THE  ISOMETRIC  SYSTEM. 

Tetartohedrism,  or  the  independent  occurrence  of 
one  qimrter  of  the  planes  of  a  holohedral  form  (p.  40), 
may  be  regarded  as  due  to  the  simultaneous  develop- 


THE  ISOMETRIC  SYSTEM. 


77 


ment  of  two  sorts  of  hemihedrism.  To  discover  what 
kinds  of  tetartohedrism  may  occur  in  any  crystal  sys- 
tem, we  may  therefore  combine  its  hemihedral  selec- 
tions in  every  way  possible  and  note  which  of  the  re- 
sults satisfy  the  required  conditions.  In  the  isometric 
system  the  result  is  the  same,  whichever  two  hemihe- 
drisms  we  combine.  This  will  be  evident  upon  an 
examination  of  the  three  adjoining  figures  (105,  106 
and  107)  representing  the  three  methods  of  hemihe- 


FIG.  105. 


FIG.  106. 


FIG.  107. 


dral  selection.     Whichever  two  of  these  three  figures 
we  imagine  superposed,  the  result  is  the  same,  viz., 
three  alternate  planes  surviving  in 
alternate  octants,  as  shown  in  Fig. 
108.    If  we  imagine  the  twelve  white 
planes  on  this  hexoctahedron  to  be 
extended  until  tjiey  mutually  inter- 
sect, the  result  will  be  a  new  form, 
bounded    by    irregular    pentagons. 
There  must,  of  course,  be  four  of 
these  quarter-forms  derivable  from 
every  hexoctahedron.     The  two  developed  from  the 
alternating  planes  of  the  same  octants  will  be  enantio- 
morphous  (Figs.  109  and  110) ;  and  to  each  of  these 
two  forms  there  will   be  a   congruent  form,  derived 


FIG.  108. 


78 


CRYSTALLOGRAPHY. 


from  the  sets  of  planes  in  the  other  octants.  The 
enantiomorphons  pairs  are  distinguished  as  right-  and 
left-handed,  and  the  congruent  pairs  are  distinguished 


Fiat*. 


FiG.tro. 


as  positive  and  negative.  These  quarter-forms  are 
called  tetrahedral-pentagonal  dodecahedrons.  They  are 
designated  as  follows  : 

Positive  right-handed,  -|  --  -A  —  r,  KK  \lkh\ 


4 
Negative  right-handed, j —  r,  KTT  \  Mh  \ 

Positive  left-handed,     -| -A —  Z,  Kit  \  klh  \ 


Negative  left-handed, j — I,  KTT  \lkh 


-  congruent 
pair. 


congruent 
pair. 


The  combination  of  a  positive  right-handed  with  a 
negative  left-handed  form,  or  vice  versa,  would  produce 
the  pentagonal  icositetrahedron  (gyroidal  form).  The 
union  of  the  members  of  either  congruent  pair  would 
produce  the  diploid  (parallel-face  form)  ;  while  the 
union  of  the  members  of  either  enantiomorphous  pair 
would  produce  the  hextetrahedron  (inclined-face  form). 

No  other  geometrically  new  form  can  result  in  the 
isometric  tetartohedrism.  This  will  be  made  clear  by 


THE  ISOMETRIC  SYSTEM. 


79 


the  following  ten  figures  (111-120)  which  show  the  result 
of  the  simultaneous  application  of  two  hemihedrisms 


FIG.  117. 


FIG.  118. 


FIG.  119. 


to  each  isometric  holohedron,  as 

well  as  the  effect  of  a  hemihe- 

drism  different  from  its  own,  upon 

every  hemihedron.     All  of  these 

forms   may,  however,  be  really 

tetartohedral  on  account  of  their 

molecular  structure,  even  when  FIG.  120. 

externally  they  do  not  differ  from  hemihedral  or  holo- 

hedral  forms. 


80 


CRYSTALLOGRAPHY. 


FIG.  121. 


There  are  two  ways  of  recognizing  a  crystal  as 
tetartohedral :  (1)  by  identifying  on 
it  the  faces  of  the  geometrically  te- 
tartohedral form  ;  or  (2)  by  discov- 
ering on  it  faces  of  forms  belonging 
to  two  different  kinds  of  hemihe- 
drism.  For  instance,  if  we  observe 
on  the  same  crystal  the  tetrahedron 
(— o)  and  the  pentagonal  dodeca- 
hedron (p),  as  may  sometimes  be 
done  in  the  case  of  sodium  chlorate  (Fig.  121),  we 
may  conclude  that  the  substance  is  tetartohedral,  and 
therefore,  that  if  the  most  gen- 
eral form  occurred  on  it,  it  could 
be  represented  by  but  one  quar- 
ter of  its  planes. 

Other  substances  showing  iso- 
metric tetartohedral  crystalliza- 
tion are :  sodium  bromate ;  thje 
nitrates  of  lead,  barium,  and 
strontium  ;  and  uraiyl  sodium 
acetate,  NaUO2(C2H3O2)3.  A 

manifold  combination  observed  on  a  crystal  of  barium 
nitrate  is  shown  in  Fig.  122.     It  has  the  forms 

ooOoo,jlOOf(a);        -£,**{  111 }  (o)  ; 


2  O2 


-        r,  K«  {214} 


and 


•,K7C\35l\(n). 


CHAPTER  IV. 

THE  TETRAGONAL  SYSTEM.* 

HOLOHEDRAL  DIVISION. 

Second  Class  of  Crystal  Systems.  The  second  class  of 
crystal  systems,  defined  on  p.  44,  comprises  all  the 
forms  which  possess  a  single  principal  plane  of  sym- 
metry. It  is  customary  to  place  this  principal  plane, 
to  which  all  of  the  secondary  planes  of  symmetry  are 
perpendicular,  in  a  horizontal  position.  In  a  physi- 
cal sense,  the  symmetry  of  the  two  systems  belonging 
to  this  class  is  the  same,  and  the  physical  properties 
of  all  of  their  crystals  are  identical.  Their  differences 
of  form  are  such  as  arise  from  the  presence  of  two 
lateral  axes  intersecting  at  90°,  or  of  three  lateral  axes 
intersecting  at  60°. 

On  account  of  its  higher  grade  of  symmetry  it  might 
seem  more  logical  to  consider  the  hexagonal  system 
first.  It  is  nevertheless  deemed  advisable  to  give  the 
tetragonal  system  precedence,  inasmuch  as  it  is  the 
easier  of  the  two  to  understand;  and,  at  the  same 
time,  the  more  closely  related  to  the  isometric  system. 

Symmetry.     The  distribution  of  the  planes  of  sym- 

*  Also  called  the  quadratic,  pyramidal,  or  quaternary  system. 

81 


oz  CRYSTALLOGRAPHY. 

metry  belonging  to  complete  or  holohedral  tetragonal 
crystals  is  represented  by  Fig.  123. 
There  is  one  principal  plane  of  sym- 
metry (placed  horizontally)  which  is 
intersected  by  four  vertical  secondary 
planes  of  symmetry.  These  latter 
meet  at  angles  of  45°,  in  a  common 
line — the  principal  axis  of  symmetry. 
Alternate  secondary  planes  of  sym- 
metry only  are  crystallographically 
interchangeable,  while  contiguous 
planes  are  not  interchangeable.  The  interchangeable 
planes  must  therefore  intersect  at  angles  of  90°;  and 
either  pair  of  these  may  be  regarded  as  determining, 
by  their  intersections  with  the  principal  plane  of  sym- 
metry, the  directions  of  the  two  lateral  axes.  Which- 
ever two  of  the  secondary  planes  of  symmetry  are  so 
regarded  are  called  the  axial  planes,  while  the  other 
two  are  then  known  as  the  intermediate  planes. 

Axes.  The  directions'  of  the  tetragonal  axes  of  ref- 
erence are  fully  determined  by  the  symmetry  of  the 
system.  There  must  be  one  principal  axis  (German, 
Hauptaxe),  normal  to  the  principal  plane  of  symmetry, 
and  therefore  vertical  in  its  position.  There  must 'be 
two  secondary  or  lateral  axes  (German,  Nebenaxen\ 
normal  to  the  principal  axis  and  to  each  other,  whose 
position  is  determined  by  whichever  pair  of  alternate 
planes  of  symmetry  have  been  selected  as  axial  planes. 
The  intersections  of  the  other  two  secondary  planes 
of  symmetry  with  the  principal  plane  determine  two 
directions  which  are  sometimes  referred  to  as  the 
intermediate  axes  (German,  Zwischenaxen}. 

The  directions  of  the  tetragonal  axes  are  therefore 


THE  TETRAGONAL  SYSTEM.  83 

the  same  as  those  of  the  isometric  system ;  their  es- 
sential distinction  from  the 

«  T~C 

latter  is,  however,  that  while 
the  lateral  axes  are  still 
equal  and  interchangeable 
with  each  other,  they  are 

no    longer    interchangeable    __fl_ 

with  the  principal  or  verti-  +a 

cal  axis.  We  must  there- 
fore designate  the  vertical 
axis  by  a  different  letter,  c, 
from  that  used  for  the  lat-  — ' c 

eral  axes,  although  the  use 

of  signs  is  the  same  as  has  been  explained  for  the 
isometric  system  (Fig.  124). 

The  Fundamental  Form  and  Axial  Ratio.  The  funda- 
mental or  ground-form  (German,  Grundform)  has  been 
defined  (p.  47)  as  composed  of  those  planes  which 
intercept  all  the  axes  at  their  unit  lengths.  In  the 
isometric  system,  the  equality  of  all  the  axes  precludes 
the  possibility  of  more  than  one  ground-form  ;  when, 
however,  as  in  the  present  case,  the  lateral  and  verti- 
cal axes  are  irrational  multiples  of  each  other,  there 
may  be  a  variety  of  fundamental  forms  in  the  same 
system,  just  as  there  may  be  a  variety  of  irrational 
inequalities  between  the  axes.  The  ratio  existing 
between  the  unit  lengths  of  the  two  unequal  axes  a 

s\ 

and  c  is  expressed  by  the  quotient  -,  where  a  is  as- 
sumed as  unity.  This  quotient  is  called  the  axial 
ratiOy  and  is  a  very  important  crystallographic  quan- 
tity. The  axial  ratios  derived  from  all  forms  occurring 
on  crystals  of  the  same  substance  under  the  same  con- 


84 


CRYSTALLOGRAPHY. 


ditions  must,  according  to  the  law  of  rationality  of  the 
indices  (p.  26),  be  rational  multiples  of  one  another, 
while  those  derived  from  forms  on  crystals  of  differ- 
ent substances  are  irrational  multiples  of  one  another. 
Thus  the  axial  ratio  becomes  a  physical  constant  for 
all  crystallized  matter  that  is  not  isometric,  and  serves 
to  identify  it  in  the  same  way  that  specific  gravity, 
hardness*  or  elasticity  does. 

It  is  important  to  determine  accurately  the  axial  ratio 

for     different     substances. 
This  is  easily  accomplished 
for    tetragonal    crystals    if 
we  know  by   measurement 
the  values  of  either  of  the 
interfacial     angles    belong- 
ing   to    the    ground-form. 
Thus  (Fig.  125)  if  we  as- 
sume the  length  of  the  lat- 
eral   axis    a    as    equal    to 
unity,  the  length   of  c,  ex- 
pressed in  terms  of  o,  is  = 
tg  6;  while  sin  b  =  cotg  A  =  cotg  £ X,  or  tg  b  —  tg  \Z  V^. 
Mere  inequality  in  the  length  of  the  intercepts  on 
the  lateral  and  vertical  axes  is  not  alone  enough  to 
constitute  a  tetragonal  form.     The  ratio  existing  be- 
tween these  unequal  intercepts  must  further  be  an 
irrational  quantity.     A  plane,  for  instance,  whose  sym- 
bol is  a  :  a  :  2a  has  an  intercept  on  one  axis  which  is 
unequal  to  those  upon  the  other  two  axes,  but  this  is 
the  plane  of  an  isometric  form  so  long  as  the  param- 
eter, 2,  is   a   rational  multiple  of  all  the  axes.      If, 
however,  the  parameter,  2,  is  only  a  rational  multiple 
of  the  vertical  axis,  but  an  irrational  multiple  of  the 


FIG.  125. 


THE  TETRAGONAL  SYSTEM.  85 

lateral  axes,  the  symbol  must  be  written  a  :  a  :  2c, 
and  the  plane  becomes  truly  tetragonal.  Hence  the 
axial  ratio  is  an  irrational  quantity  which  is  charac- 
teristic of  a  given  chemical  compound. 

When  two  or  more  tetragonal  pyramids  occur  on 
crystals  of  the  same  substance,  it  is  necessary  to  select 
one  of  them  as  the  ground-form,  and  from  this  to  cal- 
culate the  axial  ratio.  It  is  not  a  matter  of  great  impor- 
tance which  particular  pyramid  is  selected  for  this  pur- 
pose, since,  according  to  the  law  of  rational  indices,  the 
intercepts  of  all  planes  on  the  same  or  equivalent  axes 
must  be  even  multiples  of  one  another.  The  ratio 
between  the  intercepts  of  any  plane  on  dissimilar  axes 
must  therefore  always  be  the  axial  ratio,  or  some  even 
multiple  of  it,  for  the  particular  substance  to  which 
the  plane  belongs.  To  secure  uniformity  in  the  choice 
of  a  ground-form,  it  is,  however,  customary  to  select 
as  such  the  most  common  or  most  prominently  devel- 
oped pyramid,  or  else  one  that  is  distinguished  by 
some  physical  property  like  cleavage. 

Development  of  the  possible  Holohedral  Forms  in  the 
Tetragonal  System.  The  inequality  in  the  unit  lengths 
of  the  vertical  and  lateral  tetragonal  axes  signifies 
that  these  axes  are  crystallographically  dissimilar 
and  not  interchangeable — a  fact  which  is  also  directly 
derivable  from  the  symmetry  of  the  system  (p.  82). 
The  most  general  parameter  symbol  for  a  tetragonal 
form  is  therefore  Q 

nal  :  «2  :  me,        A^ 

which  is  evidently  capable  of  only  two  permutations, 
if  only  al  and  a2  are  interchangeable  : 

wa,  :  a>9  :  me    and    al  :  na^  :  me. 


86  CRYSTALLOGRAPHY. 

Hence  the  most  general  tetragonal  symbol  stands  for 
two  planes  in  each  octant,  or  sixteen  planes  in  all. 

The  most  general  index  symbol  (hid)  leads  us  to  the 
same  result,  when  only  h  and  Jc  are  interchangeable. 
With  every  possible  combination  of  signs,  represent- 
ing the  different  octants,  we  obtain 


Four  upper  octants. 

hid    m    IE 


Four  lower  octants. 

IE    hll    lu    IE 

TM     Ml     kid     M 


The  same  number  of  planes  for  the  most  general 
tetragonal  form  is  also  indicated  by  the  symmetry  of 
the  system.  From  Fig.  123  we  see  that  the  five  planes 
of  symmetry  divide  space  into  sixteen  similar  secfeits, 
so  that  any  crystal  face,  oblique  to  all  of  these  planes, 
necessitates  another  in  each  of  the  other  fifteen  secfents. 

The  other  possible  tetragonal  holohedrons  may  be 
derived  from  the  most  general  formula  nat  :  aa  :  me,  by 
assigning  limiting  values  to  the  two  parameters,  n  and 
m.  Inasmuch  as  it  is  customary  to  make  the  lesser 
lateral  intercept  equal  to  unity,  the  limiting  values  for 
n  are  one  and  infinity.  Since  m,  however,  refers  only 
to  a  single  axis,  its  value  may  vary  between  zero  and 
infinity. 

By  assigning  these  limiting  values  first  to  one  and 
then  to  both  of  the  parameters,  we  obtain  seven  and 
only  seven  possbile  tetragonal  form-types,  which,  as  in 
the  isometric  system  (p.  50),  fall  naturally  into  three 
classes,  as  follows : 

Class  I.  Forms  with  two  variable  parameters. 

1.  ra  ^  w,  general  symbol  becomes  nal  :  at  :  me. 

Class  II.  Forms  with  one  variable  parameter. 

2,  n  =  1,  general  symbol  becomes  a,  :  a,  :  me. 


THE  TETRAGONAL  SYSTEM. 


87 


3.  n  =  oo,  general  symbol  becomes  oo  ax  :  a2  :  me. 

4.  m  =  oo,  general  symbol  becomes  wc^  :  «2  :  oo  c. 
Class  III.  Forms  with  no  variable  parameter. 

5.  m  =  oo  and  TO  =  1,  general  symbol  becomes 

ax  :  a2  :  ooc. 

6.  m  =  oo  and  %  =  oo,  general  symbol  becomes 

oo  «!  :  aa  :  oo  c. 

7.  m  =  0  and  TO  =  1,  general  symbol  becomes 


a,  :  a. 


Oc. 


Although  identical  in  their  number,  these  tetragonal 
form-types  do  not  correspond  exactly  with  those  of  the 
isometric  system  in  character,  as  we  shall  see  from  a 
more  special  description  of  the  tetragonal  holohedrons. 

The  Ditetragonal  Pyramid.  The  most  general  tetrag- 
onal symbol,  nal  :  a2  :  me,  stands  for  a  double  pyra- 
mid containing  two  planes  in  each 
octant  (Fig.  126).  The  vertical  in- 
tercepts for  all  the  planes  of  this 
form  must  be  the  same,  because 
the  vertical  axis  is  not  interchange- 
able with  the  others.  This  form  is 
called  the  ditetragonal  pyramid.  Like 
the  most  general  isometric  form,  it 
has  three  kinds  of  edges  and  three 
sorts  of  solid  angles.  The  edges  which 
lie  in  the  principal  plane  of  symmetry 
are  called  based  edges  and  are  all  simi- 
lar. Those  which  connect  the  lateral  and  vertical  axes 
are  called  polar  edges,  and  are  alternately  dissimilar. 
The  faces  of  the  ditetragonal  pyramid  are  all  similar 
scalene  triangles.  Its  symbol  according  to  the  nota- 
tion of  Naumann  is  mPn,  its  general  index  symbol  is 


FIG.  126. 


CRYSTALLOGRAPHY. 


Tetragonal  Pyramids  of  the  First  and  Second  Order.  If, 
in  the  parameter  symbol  of  the  most  general  tetrag- 
onal form,  nat  :  aa  :  me,  the  parameter  n  be  given  its 
two  limiting  values,  while  the  parameter  m  is  allowed 
to  remain  unchanged,  two  new  eight-sided  pyramids 
result  which  are  the  limiting  forms  of  the  ditetragonal 
pyramid  in  two  directions. 

If  n  be  made  equal  to  unity,  the 
formula  becomes  a, :  aa :  me,  which  rep- 
resents a  form  each  of  whose  planes 
occupies  a  single  octant  (Fig.  127). 
This  is  called  the  tetragonal  pyramid 
of  the  first  order.  Its  faces  are  similar 
isosceles  triangles,  and  its  edges  and 
solid  angles  are  both  of  two  kinds. 
Its  symbols,  according  to  Naumann 
and  Miller,  are  mP  and  \hhl\. 

If,  on  the  other  hand,  n  be  given 
its  maximum  value,  infinity,  the  for- 
mula becomes  GO  a1 :  a2 :  me,  which  represents  another 
tetragonal  pyramid,  each  of  whose 
planes  is  common  to  two  octants 
(Fig.  128).  This  is  called  the 
tetragonal  pyramid  of  the  second 
order.  Its  symbols,  according  to 
Naumann  and  Miller,  are  mPoo 
and  {Ohi\. 

The  diagram,  Fig.  129,  illus- 
trates the  relations  of  the  three 
tetragonal  pyramids,  as  shown  in 
their  cross-sections.  The  inner 
square  gives  the  position  of  the 
pyramid  of  the  first  order  with  ref- 
erence to  the  axes,  and  the  outer  square  that  of  the 


FIG.  127. 


FIG.  128. 


THE  TETRAGONAL  SYSTEM. 


89 


FIG.  129. 


pyramid  of  the  second  order.  The  octagon  between 
these  two  squares  gives  the  inter- 
mediate position  of  the  ditetragonal 
pyramid.  The  pyamids  of  the  first 
and  second  orders  differ  only  in  their 
relative  positions.  The  order  of 
either  depends  entirely  on  which 
of  the  two  pairs  of  interchangeable 
axes  (p.  82)  we  select  as  lateral  axes. 

No  new  type  of  tetragonal  form  is 
obtained  by  giving  the  vertical  parameter,  ra,  the  value 
unity,  because  this  is  not  a  limiting  value  for  the 
vertical  axis.  It  is,  however,  customary  to  designate 
those  pyramids  whose  vertical  parameters  are  unity  as 
unit  pyramids.  They  do  not  in  reality  differ  from  the 
other  pyramids,  any  more  than  these  differ  among  them- 
selves, since  it  is  in  a  measure  arbitrary  which  of  the 
pyramids  belonging  to  a  substance  is  selected  as  the 
ground-form.  The  parameter  symbols  of  these  unit 
pyramids  are  written  Pn,  P,  and  P  oo  ;  and  their  indices 
\hkk},  \lll\  and  J101|. 

Tetragonal  Prisms.     If  we  give  the  lateral  parameters 
of  the  tetragonal  formula  the  values  they  possess  in 


Fia.  130.  FIG.  131.  Fia.  132. 

the  three  types  of  pyramids,  and,  at  the  same  time,  in- 


90  CRYSTALLOGRAPHY. 

crease  the  vertical  parameter,  m,  to  its  maximum  limit, 
infinity,  three  types  of  prisms  will  result,  which  cor- 
respond in  all  respects  to  the  three  pyramids,  except 
that  their  planes  are  parallel  to  the  vertical  axis  (Figs. 
130,  131,  and  132).  These  are  all  open  forms  (p.  36) 
and  they  cannot  therefore  occur  except  in  combination. 
Their  relative  positions  are  shown  in  Fig.  129.  They 
are  named,  to  accord  with  the  corresponding  pyramids : 
the  ditetragonal  prism,  <x>Pn,  {hkQ\  ;  the  tetragonal 
prism  of  the  first  order,  ooP,  jllOf  ;  and  the  tetragonal 
prism  of  the  second  order,  oo  P  oo,  j  100 } . 

The  last-named  form,  in  spite  of  its  being  called  a 
prism,  really  belongs  to  the  type  of  pinacoids  (p.  36) ; 
just  as  the  pyramid  of  the  second  order  really  belongs 
to  the  type  of  prisms,  since  its  planes  are  parallel  to 
one  of  the  axes. 

The  Basal  Pinacoid.  The  last  of  the  seven  possible 
types  of  tetragonal  forms  is  produced  by  giving  the 
vertical  parameter,  m,  its  minimum  limiting  value,  zero. 
This  causes  all  of  the  pyramidal  types  to  merge  into 
one  plane  whose  position  is  that  of  the  principal  plane 

of  symmetry  (Fig.  123).    When 
.}.<.  J       such   a  plane,  whose  parame- 

ter symbol  is  a,  :  aa  :  Oc,  is 
divided  into  the  pair  of  par- 
allel planes  necessary  to  pro- 


-a—  "t-fla   duce  a  holohedral  form  (p.  18), 

they  have  the  position  shown 
in  Fig.  133,  parallel  to  both 
lateral  axes.  These  planes  are 
of  unlimited  extent,  and  can 
hence  only  occur  in  combina- 
tion with  other  forms.  They  are  called  basal  pina- 


THE  TETRAGONAL  SYSTEM. 


91 


coids,  and  are  designated  in  the  notations  of  Naumann 
and  Miller  by  the  symbols^  OP(001 }  (p.  29). 

Limiting  Forms  in  the  Tetragonal  System.  The  rela- 
tion of  limiting  forms  among  the  tetragonal  holohe- 
drons  may  be  illustrated  by  the  following  diagram, 
where  ra  indicates  a  vertical  parameter  greater  than 


unity,  and  —  one  less  than  unity. 


OP 

I 


OP 

I 


OP       Pinacoid. 


P 

mP 


m 
Pn 

mPn 


Poo 
mPoo 


Pyramids. 


I  I 

ooP     -    -    -      ooP%     .    -    -     ooPoo        Prisms. 

Limiting  forms  within  a  single  system  are  produced 
by  variations  in  parameters,  until  these  reach  a  fixed 
or  limiting  value.  This  we  have  seen  illustrated  in 
both  the  isometric  and  tetragonal  systems,  and  we 
shall  find  that  it  is  equally  true  of  all  the  other 
systems.  There  is,  however,  another  way  in  which 
limiting  forms  may  be  produced,  and  that  is  by 
variations  of  the  axial  ratio.  In  such  cases  the  limit- 
ing forms  always  belong  to  another  system,  of  a 
higher  grade  of  symmetry  than  that  possessed  by 
the  forms  which  they  limit.  For  instance,  the  first 
of  the  two  following  figures  (134)  represents  a 
tetragonal  pyramid,  which  remains  such  as  long  as 
the  vertical  axis  is  either  greater  or  less  than  the 
lateral  axes,  no  matter  how  small  the  difference  may 


92  CRYSTALLOGRAPHY. 

be.  In  the  case  of  the  iron-copper-sulphide,  chal- 
copyrite,  the  ratio  a  :  c  is  1  :  0.9856  +>  which  is  very 
near  1:1.  If  this  limit  were  actually  reached,  how- 
ever, the  form  would  cease  to  be  tetragonal,  and 
would  become  its  limiting  form  in  the  isometric 
system — the  octahedron  (Fig.  135),  whose  interfacial 
angles  are  all  109°  28'  16".4,  instead  of  108°  40'  and 


FIG.  134. 


FIG.  135. 


109°  53'  as  they  are  in  chalcopyrite.  Many  crys- 
tals approach  so  closely  to  their  limiting  forms  in 
systems  of  a  higher  grade  of  symmetry  that  their 
true  character  can  only  be  discovered  by  an  examina- 
tion of  their  optical  properties,  or  by  some  other 
physical  test  more  delicate  than  the  measurement  of 
their  interfacial  angles.  Such  a  close  approach,  on 
the  part  of  any  crystal,  to  a  grade  of  symmetry 
higher  than  it  really  possesses  is  called  by  Tscher- 
mak  pseudo-symmetry. 

The  Crystal  Series.  The  whole  sequence  of  possible 
tetragonal  holohedral  forms,  as  above  developed, 
may  evidently  belong  to  a  single  set  of  axes.  In 
other  words,  their  nature  and  existence  is  dependent 
upon  the  symmetry  of  the  system,  but  is  not  depend- 


THE  TETRAGONAL  SYSTEM.  93 

ent  upon  the  axial  ratio.  But  such  planes  only  can 
occur  on  crystals  of  a  given  substance  as  satisfy  the  law 
of  rational  indices  for  its  particular  axial  ratio.  Inas- 
much, however,  as  there  are  as  many  distinct  axial 
ratios  in  the  tetragonal  system  as  there  are  chemical 
substances  crystallizing  with  the  tetragonal  symmetry 
(p.  82),  it  is  plain  that  each  substance  has  its  own 
tetragonal  forms  which  represent  the  same  types  as 
those  of  other  substances  andyet  which  differ  from  them 
in  their  exact  axial  inclinations  and  interfacial  angles. 

Such  a  difference  necessitates  the  introduction  of  a 
new  group  of  crystal  forms  called  the  Crystal  Series. 
This  may  be  defined  as  the  sum  of  all  the  crystal 
forms  which  are  possible  upon  the  same  set  of  axes  or 
with  the  same  axial  ratio.  Each  series  may  contain  a 
complete  representation  of  all  the  types  of  forms, 
while  the  number  of  different  series  is  equal  to  the 
number  of  different  substances  crystallizing  in  the 
system. 

In  the  isometric  system  it  is  evident  that  distinct 
crystal  series  are  impossible,  since  for  all  substances 
the  axial  ratio  is  the  same. 

Holohedral  Tetragonal  Forms  in  Combination.  The 
simpler  tetragonal  combinations  are  readily  intelli- 


Fio.  136.  Fio.  137 


94 


CRYSTALLOGRAPHY. 


gible,  as  will  be  seen  from  the  accompanying  illustra- 
tions. Fig.  136  shows  the  union  of  a  pyramid  and 
prism  of  the  same  order,  while  Fig.  137  gives  the  re- 
sult of  a  union  of  the  same  forms  belonging  to  differ- 
ent orders.  The  next  three  figures  show  combinations 
of  pyramids  of  the  first  and  second  orders— Fig.  138 


FIG.  138. 


FIG.  139. 


when  the  two  have  equal  vertical  parameters ;  Fig. 

139  when  the  pyramid  of  the  second  order  is  the 
more  obtuse,  and  Fig.  140  when 
it  is  the  more  acute  of  the  two 
forms. 

The  four  succeeding  figures 
represent  certain  more  complex 
tetragonal  combinations.  Fig. 
141  shows  a  crystal  of  manga- 
nese dioxide,  MnO3  (polianite), 

bounded  by  the  unit  pyramid  of  the  second  order, 


FIG.  140. 


FIG.  142. 


THE  TETRAGONAL  SYSTEM. 


95 


Poo,  {101}  (e),  the  ditetragonal  pyramid  3Pf,  {321}  (z), 
and  the  ditetragonal  prism  oo  P2,  {210}  (A).  Fig.  142 
represents  a  crystal  of  boron  with  the  unit  pyramids 
of  both  orders,  P,  {111}  (o)  and  Poo ,  {101}  (o'),  the 
steeper  pyramid  of  the  first  order,  2P,  \  221 }  (2o),  and 
the  prisms  of  both  orders  oo  P,  { 110 }  (m)  and  oo  P  oo, 
{100}(m').  Fig.  143  gives  the  planes  observed  on  a  crys- 
tal of  hydrous  nickel  sulphate  :  the  basal  pinacoid,  (c) ; 
three  pyramids  of  the  second  order,  (m),  (o)  and  (q) ; 
two  pyramids  of  the  first  order,  (n)  and  (p) ;  and  the 


FIG.  143. 


FIG.  144. 


prisms  of  both  first  and  second  orders,  (r)  and  (s).  Fi- 
nally, in  Fig.  144  we  have  a  very  complex  combination 
of  thirteen  tetragonal  forms  occurring  on  the  mineral 
vesuvianite :  the  prism  of  the  first  order,  (d) ;  prism 
of  the  second  order,  (M) ;  basal  pinacoid,  ( s) ;  three 
pyramids  of  the  first  order,  P,  \lll\  (c),  2P,  {221}  (6), 
and  4P,  {441}  (r) ;  two  pyramids  of  the  second  order, 
Poo,  {101}  (o)  and  2Poo ,  {201}  (u) ;  the  ditetragonal 
prism,  oo  P2,  { 210 }  (/) ;  and  the  four  ditetragonal 
pyramids,  2P2,  {121}  («),  4P4,  {141}  (a?),  4P2,  {241}  (c), 
and  fP3,{  132 }{a). 

As  other  examples  of  holohedral  tetragonal  crystal- 


96  CRYSTALLOGRAPHY. 

lization  may  be  mentioned  stannic  oxide,  SnO2  (cassit- 
erite) ;  titanium  dioxide,  TiO2,  in  two  of  its  forms,  rutile 
and  anatase ;  zircon,  ZrSiO4 ;  the  chloride,  iodide, 
and  cyanide  of  mercury;  and  the  hydrous  silicate, 
apophyllite. 


HEMIHEDRAL  DIVISION  OF  THE  TETRAGONAL  SYSTEM. 

Possible  Kinds  of  Tetragonal  Hemihedrism.  A  careful 
inspection  of  the  most  general  tetragonal  form — the 
ditetragonal  pyramid — shows  that  it  is  possible  to 
select  one  half  of  its  planes  in  three  different  ways  so 
as  to  satisfy  the  conditions  of  hemihedrism  given  on 
p.  41.  These  three  methods  of  selection  give  rise  to 
three  distinct  kinds  of  hemihedrism,  which  are  closely 


FIG.  145. 


FIG.  146. 


FIG.  147. 


analogous  to  the  three  kinds  of  isometric  hemihedrism. 
The  possible  methods  of  selection  are  represented  in 
the  three  annexed  figures  (145, 146  and  147).  They  are 
(1)  by  alternate  planes,  (2)  by  pairs  of  planes  inter- 
secting in  the  principal  plane  of  symmetry,  and  (3)  by 
alternate  octants. 

The  first  produces  forms  devoid  of  all  symmetry 
and    therefore   enantiomorphous.      It  is   called   the 


THE  TETRAGONAL  SYSTEM.  97 

trapezohedral  hemihedrism.  The  second  produces  forms 
having  one  principal,  but  no  secondary  planes  of  sym- 
metry, and  is  called  the  parallel-face  or  pyramidal 
hemihedrism.  The  third  produces  forms  having  two 
secondary,  but  no  principal  planes  of  symmetry,  and 
is  called  the  inclined-face  or  sphenoidal  hemihedrism. 

Trapezohedral  Hemihedrism,  The  extension  of  al- 
ternate planes  of  the  ditetragonal  pyramid  until 
they  intersect  produces 
an  asymmetrical  figure 
bounded  by  eight  similar 
trapeziums,  which  meet 
in  three  kinds  of  edges 
and  two  kinds  of  solid 
angles.  Two  similar  but 
enantiomorphous  forms 
of  this  kind  are  derivable  Fio.^148. 
from  every  ditetragonal  pyramid  (Figs.  148  and  149). 
They  are  called  tetragonal  trapezohedrons.  Their  gen- 
eral symbols,  according  to  Naumann  and  Miller,  are 

mPn  mPn  _ 

-r,    r\lchl\         and  -I,     rAhkl}. 

A  A 

It  is  evident  that  no  other  new  form  can  result  by 
this  method  of  selection  from  any  of  the  more  special 
tetragonal  forms,  because,  in  each  of  them,  the  planes 
correspond  to  at  least  two  contiguous  planes  of  the 
most  general  form. 

This  mode  of  crystallization  has  never  been  ob- 
served on  natural  minerals,  but  it  has  several  repre- 
sentatives among  organic  salts.  Examples  of  these  are 
sulphate  of  strychnine,  sulphate  of  ethylendiamine, 
carbonate  of  guanidine,  and  diacetylphenol  phtalline. 


98 


CRYSTALLOGRAPHY. 


Pyramidal  Hemihedrism,  The  extension  of  alter- 
nate planes  on  the  upper  half  of  the  ditetragonal 
pyramid,  and  of  those  directly  below  them  on  the 
lower  half,  as  indicated  in  Fig.  146,  p.  96,  must  pro- 
duce a  tetragonal  pyramid,  which  is  in  all  respects 
like  that  of  the  first  or  second  order,  except  in  its 
position.  While  the  two  holohedral  tetragonal  pyra- 
mids differ  45°  in  their  positions,  this  hemihedral 
tetragonal  pyramid,  called  the  pyramid  of  the  third 
order,  occupies  an  intermediate  place,  which  is  deter- 
mined by  the  exact  parameters  of  the  ditetragonal 
pyramid  from  which  it  is  de- 
rived. This  may  be  best  illus- 
trated by  the  adjoining  diagram 
(Fig.  150),  which  shows  the  rela- 
tive positions  of  the  tetragonal 
pyramids  in  cross-section,  like 
Fig.  129,  except  that  the  alter- 
nate faces  of  the  ditetragonal 
FIG.  150.  form  have  here  been  extended  to 

intersection.  The  new  square  thus  formed  represents 
the  cross-section  of  the  pyramid  of  the  third  order. 
The  symbols  of  the  corresponding  half-forms  deriv- 
able from  any  ditetragonal  pyramid  are 


\~mPn~] 
~2J' 


ir\lM\ 


and         — 


2' 


7f\hE}. 


The  same  modification  is  produced  by  this  method 
of  hemihedral  selection  upon  the  ditetragonal  prism, 
since  each  of  its  planes  corresponds  to  the  pair  of 
planes  selected  on  the  most  general  form.  Thus  two 
hemihedral  prisms  of  the  third  order  result  from  each 


THE  TETRAGONAL  SYSTEM.  99 

ditetragonal  prism,  whose  intermediate  positions  are 
likewise  shown  by  Fig.  150.     Their  symbols  are 


If  we  compare  the  other  tetragonal  holohedrons  with 
the  most  general  form,  we  shall  readily  see  that  each 
of  their  planes  corresponds  to  two  contiguous  planes 
in  the  upper  half  of  the  crystal,  and  hence  that  both 
pyramids  and  prisms  of  the  first  and  second  order 
must  reproduce  themselves  in  the  pyramidal  hemihe- 
drism  without  geometrical  alteration.  Two  new  forms 
only  are  possible  by  parallel-face  hemihedrism  in  the 
tetragonal  system,  as  was  also  found  to  be  the  case 
in  the  isometric  system. 

The  tetragonal  forms  of  the  third  order  manifest 
their  true  character  in  combination  with  other  forms, 
as  may  be  seen  in  the  following  examples.  Fig.  151 
shows  a  crystal  of  lead  tungstate,  PbWO4  (stolzite), 
bounded  by  the  unit  pyramid  of  the  first  order, 
P,  jlllj  (o),  and  by  the  prism  of  the  third  order, 


^  {430}  (p).      Fig.  152  represents  a  crys- 

tal of  yttrium  niobate  (fergusonite)  with  the  basal 
pinacoid,  OP,  J001J  (c)  ;  the  unit  pyramid,  P,  {111}  (s); 
and  the  pyramid  and  prism  of  the  third  order, 


(»)      and       -  ,  *{230)  (r). 


Fig.  153  shows  a  combination  of  forms  sometimes  ob- 
served on  the  silicate  scapolite,  with  the  unit  pyra- 
mid, P,  j  111  }  (o)  ;  the  prisms  of  the  first  and  second 


100  CRYSTALLOGRAPHY. 

orders,  oo  P,  J110}  (M )  and  oo  P  oo,  jlOO}  (6);  and  the 

po  pQ— I 

pyramid  of  the  third  order,  -f-    -g-  L  ^{311}  (s). 


FIG.  151. 


FIG.  152. 


FIG.  153. 


Other  examples  of  tetragonal  substances  showing 
pyramidal  hemihedrism  are  calcium  tungstate,  GaWO4 
(scheelite) ;  lead  molybdate,  PbMoO4  (wulfenite); 
hydrous  magnesium  borate,  MgB2O4  -f-  3aq  (pinnoite)  ; 
erythroglucine  (C4H10O4)  and  toluosulphanide. 

Sphenoidal  Hemihedrism.  The  selection  of  planes 
by  alternate  octants,  as  indicated  by  Fig.  147  on  p.  96 
for  the  sphenoidal  hemihedrism,  can  produce  only  two 
geometrically  new  forms  in  the  tetragonal  system, 
since  on  only  two  holohedrons 
of  this  system  do  the  planes 
belong  exclusively  to  a  single 
octant.  These  two  forms  are  the 
ditetragonal  pyramid  and  the 
pyramid  of  the  first  order. 

The   result   of  extending  the 
pairs  of  planes  occupying  alter- 
nate octants  on  the  most  general 
FIG.  154.  form    until     they    intersect    is 

shown  in   Fig.  154.     The   figure    thus    produced    is 


THE  TETRAGONAL  SYSTEM.  101 

bounded  by  eight  similar  scalene  triangles,  intersect- 
ing in  three  sorts  of  edges  and  two  sorts  of  solid 
angles.  It  is  called  the  tetragonal  scalenohedron.  The 
two  forms  resulting  from  the  extension  of  the  two 
halves  of  the  ditetragonal  planes  are  similar  and 
congruent  (p.  67).  They  may  be  made  to  coincide  by 
revolving  either  one  through  90°  about  its  vertical  axis. 

Their  symbols  are  +  —   —  ,  K\liH\  and  --  ~—  ,  K{hJcl}. 

A  A 

The  survival  of  alternate  planes  on  the  pyramid 
of  the  first  order  is  analogous  with 
that  of  the  corresponding  planes  on 
the  isometric  octahedron  to  produce 
the  tetrahedron  (p.  73).  The  result 
is  a  form  bounded  by  four  isosceles 
triangles,  intersecting  in  two  sorts 
of  edges  and  with  one  kind  of  solid 
angles  (Fig.  155).  This  is  called  the 
tetragonal  sphenoid,  and  the  symbols  of 
the  two  congruent  half-  forms  are  FIG.  155. 

and          - 


The  sphenoid  is  well  calculated  to  exhibit  the  in- 
clined-face character  of  this  hemihedrism.  There  are 
acute  and  obtuse  sphenoids,  according  as  the  vertical 
axis  is  longer  or  shorter  than  the  lateral  axes,  and 
between  these  two  classes  the  isometric  tetrahedron 
stands  as  a  limiting  form  of  each. 

These  two  new  sphenoidal  hemihedrons  have  lost 
the  three  axial  planes  of  symmetry  belonging  to  the 
tetragonal  system  (Fig.  123),  but  they  retain  the  two 
intermediate  planes  of  symmetry. 


102 


CRYSTALLOGRAPHY. 


All  other  tetragonal  forms  must  appear  on  crystals 
of  sphenoidal  substances  as  apparent  holohedrons, 
because  they  are  incapable  of  geometrical  modifica- 
tion by  this  method  of  selection. 

As  examples  of  sphenoidal  crystallization  may  be 
cited  the  iron-copper-sulphide,  FeSaCu  (chalcopyrite), 


Fio.  156. 


Fio.  157. 


(Fig.  156,  ±        ,  and  urea  (CH4NaO)  (Fig.  157,  OP  (c), 
P 


Tetartohedrism  in  the  Tetragonal  System.  The  simul- 
taneous occurrence  of  two  different  kinds  of  tetragonal 
hemihedrism  is  theoretically  capable  of  producing 
tetartohedral  forms,  which  may  at  any  time  be  found 
on  natural  crystals.  A  consideration  of  Figs.  145,  146, 
and  147  (p.  96)  will  show  that  the  superposition  of  the 
last  upon  either  of  the  other  two  will  produce  a  sur- 
vival of  one  quarter  of  the  holohedral  planes  in  such 
a  manner  as  to  satisfy  the  conditions  of  tetartohedrism 
(p.  41)  ;  while  the  superposition  of  the  second  of  these 
figures  upon  the  first  will  produce  hemimorphism  in 
the  direction  of  the  vertical  axis. 


THE  HEXAGONAL  SYSTEM. 


105 


FIG.  159. 


system  only  in  having  six  vertical  secondary  planes, 
meeting  at  angles  of  30°  (Fig.  159),  instead  of  four 
secondary  planes  meeting  at  angles  of  45°  (Fig. 
123,  p.  82).  Here  also  only  alter- 
nate secondary  planes  are  crystal- 
lographically  equivalent  and  inter- 
changeable, so  that  we  have  again 
two  sets  of  three  axial  and  three  inter- 
mediate vertical  planes  of  symmetry. 
The  principal  ^ind  axial  planes  of 
symmetry  together  divide  space  into 
twelve  similar  wedge-shaped  secants, 
called  dodecants,  which  are  analogous 
to  the  isometric  and  tetragonal  octants. 

Axes.  The  directions  of  the  hexagonal  axes  of  ref- 
erence are  determined  by  the  planes  of  symmetry, 
just  as  they  are  in  the  tetragonal  system.  The  prin- 
cipal axis  of  symme- 
try (c)  is  employed  as 
the  vertical  axis,  while 
the  intersections  of 
5*  either  of  the  two  sets 

^ — ~-^+fla  of  secondary  planes 
of  symmetry  with  the 
principal  plane  give 
three  equal  lateral 
axes,  all  normal  to 
the  principal  axis  and 
inclined  60°  and  120°  to  one  another.  It  is  custom- 
ary to  designate  the  equal  lateral  axes  by  the  letters 
a,,  a2  and  «3,  in  the  order  indicated  in  Fig.  160;  and 
to  assign  to  their  extremities,  alternately,  plus  and 


-fc 


FIG.  160. 


106 


CRYSTALLOGRAPHY. 


±a» 


minus  signs,  as  first  suggested  by  Bravais.*  The  in- 
tersections of  the  intermediate  planes  of  symmetry 
with  the  principal  plane  give  three  additional  direc- 
tions which  bisect  the  angles  between  the  lateral  axes 
and  may  be  called  the  intermediate  axes.  Their  posi- 
tion is  shown  by  the  dotted  lines  in  Fig.  160. 

The  Fundamental  Form  and  Axial  Ratio.     The  signifi- 
cance of  these  terms  has  already  been  fully  explained 
,  a  _flj  in  speaking  of  the  tetrag- 

>!  /  onal   system.     The   hex- 

agonal fundamental  or 
ground-form  is  bounded 
by  planes  which  inter- 
sect the  principal  and 
two  contiguous  lateral 
axes  at  their  unit  lengths. 
This  necessitates  each 
plane  being  parallel  to 
FlG- 161-  the  remaining  lateral 

axis,  as  will  be  clear  from  the  adjoining  cross-section 
of  the  ground-form  (Fig.  161).  The  parameter  sym- 
bol of  such  a  form  must  therefore  be  al  :  a2  :  oo  a3  :  c ; 
and  its  index  symbol,  { 1101 } .  The  indices  of  the  twelve 
planes  bounding  the  complete  form,  designated  by 
their  particular  signs,  are 

(above)     1011    Olll    1101    1011    Olll    1101 
(below)     1011    Olll    1101    1011    Olll    1101 

*  Other  authors  have  employed  other  sets  of  axes  for  the  hexago- 
nal system.  Schrauf  used  only  three,  at  right  angles,  the  ratio  be- 
tween the  two  lateral  axes  being  1  :  V3.  This  he  called  the  ortfio- 
Jiexagonal  system.  Miller  used  three  axes  parallel  to  the  edges  of 
the  fundamental  rhombohedron. 


THE  HEXAGONAL  SYSTEM. 


107 


The  axial  ratio  for  a  given  substance  may  be  calcu- 
lated from  any  hexagonal 
pyramid  that  is  assumed 
as  its  ground-form,  just 
as  in  the  tetragonal  sys- 
tem. The  only  difference 
between  Fig.  162  and  Fig. 
125  (p.  84)  is  that  the 
side  a  of  the  spherical 
triangle  is  60°  instead  of 
45°.  The  value  of  the 
axis  c,  in  terms  of  the  lat- 
eral axis  a,  is  equal  to  tg 

sin  b  =  cotg  A  1/3,  where  B  is  one  half  the  basal  edge 
of  the  pyramid,  and  A  one  half  its  polar  edge. 

Development  of  the  possible  Holohedral  Forms  in  the 
Hexagonal  System.  This  is  strictly  analogous  to  the 
development  of  holohedral  tetragonal  forms  (p.  85). 
The  most  general  parameter  formula  is 


Fl°-  162- 
;  while  tg  b  =  tg  B  |/|,  or 


pa3 


me 


but,  inasmuch  as  the  three  lateral  axes  are  fixed  in  their 
mutual  inclinations  at  60°,  the  intersection  of  two  of 
them  by  any  plane,  at  distances  n  and  1  from  the  centre, 
determines  the  point  of  intersection  on  the  third  axis 


n 


by  the  same  plane,  as Tfrom  the  centre.* 

n  —  1 

the  most  general  hexagonal  formula  becomes 


Hence 


n 


na.  :  a.  :   ^  a,  :  me. 

9     n  —  1 


*  For  proof  of  this  see  Klein's  Einleitung  in  die  Kiystallberech- 
nung,  p.  319  (1870). 


108 


CRYSTALLOGRAPHY. 


which,  like  the  most  general  tetragonal  formula,  con- 
tains only  the  two  variables  m  and  n. 

The  limiting  values  for  the  vertical  parameter  m  are 
zero  and  infinity,  as  in  the  tetragonal  system  (p.  86).  If 
the  shortest  of  the  three  lateral  intercepts  be  assumed 
as  unity  and  the  intermediate  one 
be  designated  by  n,  then  the  limit- 
ing values  for  the  variable  lateral 
parameters  must  be  n  >  1  and  <  2, 

n 

and  —  — =  >  2  and  <  oo .     An  in- 
vi  —  J. 

spection  of  Fig.  163  will  make  this 
clear.  The  shortest  of  any  three 
finite  intercepts  for  the  same  plane 
must  be  intermediate  in  its  posi- 
tion between  the  other  two.  If  we 
vary  the  position  of  any  plane 
about  its  shortest  intercept  (here  --  aa),  so  that  its 
intercepts  on  the  other  axes  receive  different  values ; 

then,  when  n  =  2, r  =  2  also ;  if  now  n  be  dimin- 

71  —  JL 


-a. 


FIG.  163. 


ished, 


-1 

when  n  =  1, 


will  be  proportionately  increased,  until 


n 


-1 


=  00. 


From  the  foregoing  we  see  that,  without  the  use  of 
signs,  only  two  permutations  of  the  most  general  for- 
mula are  possible  : 

n  n 

net.  '.  OL  '. =•  GL  I  mc    and =•  a.  ;  CL  i  no,.  '.  me. 

n  —  I  n  —  1    ] 

These  correspond  exactly  to  the  two  permutations 
of  the  most  general  formula  possible  in  the  tetragonal 
system  (p.  85).  They  indicate  that  the  most  general 


THE  HEXAGONAL  SYSTEM.  109 

hexagonal  form  contains  but  two  planes  in  a  dode- 
cant,  and  that  it  is  therefore  bounded  by  twenty-four 
faces.  This  is  also  the  number  of  secants  into  which 
all  of  the  holohedral  planes  of  symmetry  divide  space 
(p.  105). 

By  assigning  limiting  values  to  one  or  both  of  the 
parameters  we  find  that  there  are  seven  types  of  hex- 
agonal holohedrons  which  are  entirely  similar  to  those 
of  the  tetragonal  system  (p.  86). 
Class  I.  Forms  with  two  variable  parameters. 

1.  m  ^  n,  general  symbol  becomes 

n 
nai  '  <*>*  '  ^^  a*  -  me- 

Class  II.  Forms  with  one  variable  parameter. 

2.  n  =  1,  general  symbol  becomes  a,  :  a2  :  oo  aa  :  me. 

3.  n  =  2,  general  symbol  becomes  2a,  :  aa  :  2a9  :  me. 

4.  m  =  oo,  general  symbol  becomes 

n 
nal  :  aa :  —• — =-  a8  :  oo  c. 

Class  III.  Forms  with  no  variable  parameter. 

5.  m  =  oo  and  n  =1,  general  symbol  becomes 

ax  :  aa  :  oo  ag  :  oo  c. 

6.  m  =  oo  and  n  =  2,  general  symbol  becomes 

2at  :  aa :  2a3  :  oo  c. 

7.  m  —  0  and  n  =  1,  general  symbol  becomes 

ax  :  aa  :  oo  a3  :  Oc. 

The  Dihexagonal  Pyramid.  The  two  planes  of  the 
most  general  hexagonal  form  belonging  to  each  dode- 
cant  combine  to  form  a  double  pyramid  bounded  by 


110  CRYSTALLOGRAPHY. 

twenty-four  similar  scalene  triangles.  It  is  entirely 
analogous  to  the  most  general  tetrag- 
onal form  (p.  87),  and  is  therefore 
called  the  dihexagoncd  pyramid  (Tig. 
164).  This  pyramid  has  three  sorts  of 
solid  angles  and  three  sorts  of  edges. 
Its  polar  edges  must  always  be 
alternately  dissimilar,  because  that 
particular  case  where  they  would  be 
FIG  164  equal  involves  the  irrational  parameter 

sin  75°.  y%  =  1.36666  +,  and  is  there- 
fore crystallographically  impossible. 

Inasmuch  as  one  of  the  two  variable  lateral  param- 
eters, n  and  -  — :=-,  always  determines  the  value  of 

the  other,  it  is  only  necessary  to  write  one  of  them  in 
the  abbreviated  symbol.  The  usage  of  Naumann  is  to 
write  the  smaller  of  the  two,  w,  so  that  his  symbol  for 
the  dihexagonal  pyramid  is  mPn,  like  that  for  the  di- 
tetragonal  pyramid,  except  that  n  can  here  never  have 
a  value  greater  than  2. 

If  we  employ  the  designation  of  the  hexagonal  axes 
suggested  by  Bravais  (p.  106),  the  most  general  expres- 
sion for  the  indices  of  any  plane  becomes  (hiJd),  in 
which  the  three  first  values  refer  to  the  lateral  axes. 
At  least  one  of  these  three  lateral  indices  must  in  every 
case  be  negative,  and  their  algebraic  sum  is  always 
equal  to  zero,  h  +  i  +  &  =  0.* 

There  is  a  difference  in  the  usage, of  different  authors 
as  to  which  particular  index  (the  largest,  medium,  or 
smallest)  a  particular  letter,  Jit  i  or  k,  represents.  We 

*  For  demonstration  of  this,  see  Groth's  Physikalische  Krystallo- 
graphic,  2dEd.,  p.  316. 


THE  HEXAGONAL  SYSTEM.  Ill 

shall  follow  the  usage  of  Groth  in  designating  the 
numerically  largest  index  (corresponding  to  the  short- 
est intercept)  by  the  letter  h, 
the  medium  index  by  k,  and 
the  smallest  by  i.  Hence, 
without  regard  to  signs, 
2k>h>k>i.  (Fig.  165.) 

The  order  in  which  the  in- 
dices are  written  in  the  sym- 
bol of  a  particular  plane  is  always  that  given  above  for 
the  axes,  al  :  aa  :  as  :  c.  Whichever  of  the  lateral  indices 
(A,  i  or  k)  refers  to  the  axis  a,  must  be  written  first,  etc. 
Suppose,  for  example  (Fig.  165),  a  plane  cuts  the  three 
lateral  axes  with  the  parameters  f  at  :  3aa  :  —  a3 ;  then 
its  indices,  or  the  reciprocals  of  the  parameters,  become 
f ,  ^,  1  or  213,  which  corresponds  to  the  order  kih. 

The  index  I,  referring  to  the  vertical  axis,  is  invari- 
ably written  last ;  thus  the  general  index  symbol  for 
the  dihexagonal  pyramid  becomes  \TdTd\  or  \htTd], 

where  h  =  —  (1-f-  k) ;  n  =  -r  and  m  =  j. 

From  what  has  just  been  stated  it  will  be  clear  that 
the  general  index  symbols  for  the  twelve  upper  planes 
of  a  dihexagonal  pyramid,  commencing  with  the  front 
dodecant  and  passing  around  to  the  left,  must  be 
kiJd        hkil        ihJd        kihl        hkil        ihkl 
hikl        khil        ikhl        Jiikl        khil        ikhl 

The  indices  of  the  twelve  lower  planes  will  be  the 
same  with  a  negative  sign  over  each  I. 

Hexagonal  Pyramids  of  the  First  and  Second  Orders. 
These  are  obtained  just  as  in  the  tetragonal  system 
by  giving  the  limiting  values,  1  and  2,  to  the  lateral 
parameter  n  of  the  general  formula. 


112 


CRYSTALLOGRAPHY. 


—a 


The  first  of  these  values  yields  the  parameter  sym- 
bol aa  :  aa  :  oo  a3  :  me,  which  stands  for  a  double  hex- 
agonal pyramid  (Fig.  166).  This 
form  is  bounded  by  twelve  similar 
isosceles  triangles,  each  of  which 
occupies  one  dodecant  and  there- 
fore corresponds  to  two  contigu- 
ous  faces  of  the  most  general 
form.  There  are  two  kinds  of 
edges,  basal  and  polar,  and  two 
kinds  of  solid  angles.  The  lateral 
axes  terminate  in  the  solid  angles, 
which  determines  this  pyramid, 
like  its  corresponding  tetragonal  form  (p.  88),  to  be  of 
the  first  order.  Its  general  abbreviated  parameter 
and  index  symbols  are  mP  and  \h07d\.  For  the 
special  case  of  a  hexagonal  pyramid  of  the  first  order 
whose  vertical  parameter  is  unity,  we  have  the  ground- 
form  of  the  system  (p.  106)  whose  symbols  are  P  and 


The  assigning  of  the  maxi- 
mum limit  to  n  produces  the 
parameter  symbol 

2a,  :  a,  :  2a8  :  me, 

which  represents   a    hexagonal 

pyramid  identical  with  that  last 

described    in    all    respects  but 

position  (Fig.  167).     The  lateral 

axes  here  terminate,  not  in  the 

solid  angles,  but  in  the  centers 

of  the  lateral  or  basal  edges,  which  marks  the  form  as 

one  of  the  second  order.     Its  symbols  are  mP2  and 


FIG.  107. 


THE  HEXAGONAL  SYSTEM. 


113 


J-&&H]  ;   or,  for   the  special  case  where   the  vertical 
parameter  is  unity,  P2  and  [2££&}.  [ff*AJ 

The  diagram  (Fig.  168)  is  intended  to  exhibit  the 
relative  positions  of  the  three  hexagonal  pyramids  in 
cross-section.  The  inner  hexagon 
corresponds  to  first-order  forms, 
and  the  outer  one  to  second-order 
forms,  while  the  intermediate  do- 
decagon represents  the  position 
of  dihexagonal  forms  (cf.  Fig.  129, 
p.  89).  The  values  of  the  first- 
and  second-order  forms  would,  of 
course,  be  reversed  if  we  chose 
to  select  the  set  of  intermediate  FIG.  IBS. 

axes  (dotted  lines)  as  axes  of  reference. 

Hexagonal  Prisms  and  Basal  Pinacoid.  These  are  so 
strictly  analogous  to  the  corresponding  forms  in  the 
tetragonal  system  that  they  require  but  a  word  of  ex- 
planation. The  three  prisms  are  derived  from  the 
three  possible  types  of  pyramids,  by  giving  its  maxi- 
mum value,  infinity,  to  the  vertical  parameter.  (Figs. 
169,  170,  and  171.) 


FIG.  169. 


FIG.  170. 


FIG.  171. 


These  forms  are  all  open,  i.e.  do  not  of  themselves 
enclose  space,  and  are  here  represented  as  of  indefi- 


114 


CRYSTALLOGRAPHY. 


nite  extent,  instead  of  in  combination  with  the  basal 
pinacoid,  as  are  the  corresponding  figures  (130,  131, 
and  132,  p.  89)  of  the  tetragonal  system.  The  names 
and  symbols  of  these  prisms  agree  with  those  of  the 
pyramids  from  which  they  are  derived :  dihexagonal 
prism  (Fig.  169),  oo  Pn,  \hikO\  ;  hexagonal  prism  of  the 
first  order  (Fig.  170),  oo  P,  JIOIOJ ;  hexagonal  prism  of 
the  second  order  (Fig.  171),  oo  P2,  J2110}. 

The  hexagonal  basal  pinacoid  (Fig.  172)  is  quite  the 
same  form  as  its  tetragonal  ana- 
logue (Fig.  133,  p.  90).  Its  sym- 
bols are  OP,  |0001|. 

Limiting  Forms  in  the  Hexagonal 
System.  The  complete  corre- 
spondence between  the  tetragonal 
and  hexagonal  systems  is  further 
illustrated  by  the  following  dia- 

FlO.  172.  ,      ,      •'  ...  ,  T 

gram  of  the  limiting  forms.     It 

differs  from  that  given  on  p.  91  only  in  the  fact  that  the 
maximum  limit  for  the  lateral  parameter  is  two  instead 
of  infinity. 


OP 

i 


IP 

m 


mP 


OP 

I 


OP        Pinacoid. 


l-Pn 

m 

Pn 
mPn 


m 


P2 

mP2 


Pyramids. 


oo  P      -     -     -     oo  Pn    -     -     -     oo  P2       Prisms. 

Holohedral    Hexagonal   Forms    in    Combination.     The 
simpler  hexagonal  combinations  are  as  readily  intelli- 


THE  HEXAGONAL  SYSTEM. 


115 


gible  as  those  in  the  tetragonal  system.  Four  of  the 
seven  possible  holohedral  types  are  open  forms  and 
therefore  cannot  occur  uncombined. 

The  prisms  may  be  closed  by  either  the  basal  pina- 
coid  (Fig.  173);  by  a  pyramid  (Fig.  174);  or  by  both 


i 

~L 


FIG.  173. 


FIG.  174. 


together.  A  hexagonal  prism  truncates  the  basal 
edges  of  a  pyramid  of  the  same  order  (Fig.  174),  and 
the  basal  angles  of  a  pyramid  of  the  opposite  order 
(Fig.  175).  (Of.  Figs.  136  and  137,  p.  93.)  A  pyramid 


FIG.  175. 


FIG.  176. 


of  the  first  order  has  its  polar  edges  truncated  by  the 
faces  of  a  pyramid  of  the  second  order  whose  vertical 
parameter  is  equal  to  its  own  (Fig.  176),  just  as  in  the 
tetragonal  system  (cf.  Fig.  138,  p.  94).  The  polar 
angles  of  any  pyramid  are  bevelled  by  the  faces  of  a 


116  CRYSTALLOGRAPHY. 

more  obtuse  pyramid  (Figs.  177  and  178) ;  or  its  basal 


FIG.  177. 


FIG.  178. 


m 


m 


edges  are  bevelled  by  the  planes  of  a  more  acute 
pyramid  of  the  same  order. 

A  more  complex  hexagonal  combination  is  repre- 
sented in  Fig.  179,  as  it  is  some- 
times observed  on  crystals  of  beryl. 
This  shows  the  basal  pinacoid, 
OP,  {0001}  (c);  the  prism  of  the  first 
order,  oo  P,  {1010}  (m) ;  two  pyramids 
of  the  first  order,  P,  {1011}  (o)  and 
2P,  {2021}  (o2) ;  a  pyramid  of  the  sec- 
ond order,  2P2,  {1121}  (q) ;  and  a  di- 
hexagonal  pyramid,  3PJ,  {3211}  («). 

The  number  of  substances  known 
to  exhibit  holohedral  hexagonal  crys- 
tallization is  very  small.  Among  them  may  be  men- 
tioned the  elements  magnesium,  beryllium,  zinc,  cad- 
mium ;  the  beryllium  silicate,  beryl  (emerald) ;  the 
antimono-silicate  of  manganese  and  iron,  Langba- 
nite  ;  pyrrhotite  (magnetic  pyrites,  Fe7S8) ;  and,  at  high 
temperatures,  tridymite,  SiO2. 

Partial  Hexagonal  Forms.  Partial  crystal  forms  pro- 
duced by  hemihedrism,  tetartohedrism,  and  hemimor- 
phism  attain  their  maximum  importance  in  the  hex- 


FIG.  179. 


THE  HEXAGONAL  SYSTEM.  117 

agonal  system.  As  has  been  just  remarked,  only  a 
very  small  proportion  of  hexagonal  substances  possess 
the  complete  holohedral  symmetry,  and  hence  the 
various  subdivisions  of  partial  forms  are  here  deserv- 
ing of  particular  attention.  One  of  these,  indeed — the 
rhombohedral — has,  on  account  of  its  extremely  fre- 
quent occurrence,  been  regarded  by  many  authors  as 
a  distinct  system. 

HEMIHEDBAL  DIVISION  OF  THE  HEXAGONAL  SYSTEM. 

Possible  Kinds  of  Hexagonal  Hemihedrism.  There  are 
three  ways  in  which  one  half  of  the  planes  of  the  di- 
hexagonal  pyramid  may  be  selected  so  as  to  satisfy 
the  conditions  of  hemihedrism.  These  three  ways  cor- 
respond precisely  with  the  three  methods  of  hemihe- 
dral  selection  applied  to  the  ditetragonal  pyramid,  as 


FIQ.  180.  FIG.  181.  FIG.  183. 

may  be  seen  by  comparing  Figs.  180,  181,  and  182  with 
Figs.  145,  146,  and  147  on  p.  96. 

The  first  method  of  selection  is  by  alternate  planes 
of  the  most  general  form  (Fig.  180).  This  produces 
new  forms  devoid  of  symmetry,  and  is  called,  as  in  the 
preceding  system,  trapezohedrdl  hemihedrism.  The  sec- 
ond method  of  selection  is  by  alternate  pairs  of  planes 
intersecting  in  the  basal  edges  (Fig.  181).  It  produces 


118 


CRYSTALLOGRAPHY. 


new  forms  with  one  principal,  but  no  secondary  planes 
of  symmetry,  and  is  called,  as  before,  pyramidal  hemi- 
hedrism.  The  third  method  of  selection  is  by  alternate 
dodecants  (Fig.  182).  This  destroys  all  but  the  three 
intermediate  secondary  planes  of  symmetry,  and  is 
called  rhombohedral  hemihedrism. 

Trapezohedral  Hemihedrism.  The  extension  of  alter- 
nate planes  of  the  dihexagonal  pyramid  until  they 
intersect  produces  a  new  half -form  bounded  by  twelve 
similar  trapeziums.  The  two  figures  resulting  from 
the  two  sets  of  alternating  planes  are  devoid  of  sym- 


Fio.  183. 


FIG.  184. 


metry,  and  are  therefore  enantiomorphous  (Figs.  183 
and  184).  They  are  called  hexagonal  trapezohedrons  and 
are  distinguished  as  right-  and  left-handed  like  their 
tetragonal  analogues  (p.  97).  Their  symbols  are  : 

mPn  <7-i7)  -i         mPn1        (7-577, 

—  r,     T\hhl};         and          —  ^—  I,     r{hikl\. 


It  is  evident  that  none  of  the  other  hexagonal  holo- 
hedrons  can  yield  a  geometrically  new  form  by  this 
method  of  selection,  because  each  of  their  faces  corre- 
sponds to  at  least  two  contiguous  faces  of  the  general 
form. 

No   example    of   trapezohedral    hemihedrism    has 


THE  HEXAGONAL  SYSTEM. 


119 


been  observed  in  the  hexagonal  system,  with  the  pos- 
sible exception  of  trie  thy  1-trimesitate,  C6H8(CO2.C2H6)3. 
Apparently  holohedral  crystals  of  this  substance  show 
slight  variations  in  their  alternating  angles,  which 
may  be  accounted  for  on  the  supposition  that  they 
are  hexagonal  trapezohedrons  with  very  large  indices. 
They  may,  however,  also  be  regarded  as  pseudohex- 
agonal  crystals  which  very  nearly  approach  hexagonal 
limiting  forms. 

Pyramidal  Hemihedrism.  The  extension  of  the  alter- 
nate pairs  of  planes  on  the  dihexagonal  pyramid 
which  intersect  in  the  basal 
edges,  produces  a  hexagonal 
pyramid  that  differs  from  the 
holohedral  pyramids  of  the  first 
and  second  order  only  in  its 
position.  This  is  intermediate  / 
between  the  positions  of  the 
other  two  forms,  since  the  axes 
terminate  neither  in  the  basal 
angle*  nor  at  the  centers  of  the 

basal  edges,  but  at  some  point  in  the  latter,  on  one 
side  of  the  center.  This  form  is  called  the  hexagonal 
pyramid  of  the  third  order.  Its  symbols  are  : 


, 


7r\Jcihi\  ;        and 


The  position  of  this  pyramid,  relative  to  the  two  other 
hexagonal  pyramids,  is  shown  in  the  cross-section, 
Fig.  185,  with  which  Fig.  168,  p.  113,  should  be  com- 
pared. 

The  planes  of  the  dihexagonal  prism  correspond  to 
the  pairs  of  planes  which  alternately  disappear  from 


120 


CRYSTALLOGRAPHY. 


the  dihexagonal  pyramid  in  order  to  produce  the  pyramid 
of  the  third  order.  This  prism  must 
therefore  be  capable  of  producing,  by 
this  method  of  selection,  two  corre- 
sponding pn'sras  of  the  third  order  (Fig. 
186)  which  are  related  to  the  prism  of 
the  first  and  second  order  in  the  same 
way  that  the  pyramids  of  the  third  or- 
der are  related  to  the  other  hexagonal 
FIG.  186.  pyramids  (Fig.  185).  The  shortened 

parameter  and  index  symbols  of  these  prisms  are  : 

'ooPrTl         "'Ini-          d        _  f00  ^n 

The  pyramidal  hemihedrism  is  also  parallel-face 
(p.  42),  as  in  the  tetragonal  system.  The  forms  of  the 
third  order  have  but  one  plane  of  symmetry,  which  is 
the  horizontal  or  principal  plane.  The  two  half-forms 
derivable  from  the  same  holohedron  are  therefore 
congruent,  and  one  may  be  brought  into  exactly  the 
position  of  the  other  by  a  revolution  about  its  vertical 
axis  through  an  angle  depending  on  the  value  of  the 
lateral  parameter,  n. 

None  of  the  other  hexagonal  holohedrons  are  ca- 
pable of  producing  geometrically  new 
forms  by  the  pyramidal  hemihedrism, 
for  the— reasons  already  stated  for  the 
analogous  tetragonal  forms  on  p.  99. 
These  will,  however,  be  made  clearer  by 
an  inspection  of  the  four  following 
figures  (187,  188,  189,  and  190),  which 
represent  the  hexagonal  forms  of  the 
first  and  second  order  with  their  faces  FIG.  m 


THE  HEXAGONAL  SYSTEM. 


121 


shaded  to  correspond  to  the  pyramidal   method   of 
selection.     On  each  figure   a  portion  of  every  face 


Fia.  188. 


FIG.  189. 


FIG.  190. 


survives,  which  is  enough  by  its  extension  to  repro- 
duce the  form  entire. 

Fig.  191  represents  a  com- 
plex combination  of  hexago- 
nal hemihedral  forms  pro- 
duced by  the  pyramidal  se- 
lection. They  are  observed 
on  crystals  of  calcium  phos- 
phate,  apatite  (Ca6Cl(PO4)3). 


FIG.  191. 

The   nine  forms   com- 
prise the  basal  pinacoid,  OP,   {  0001  }  (P)  ;   the  prism 
of  the  first  order,  GO  P,  j  1010  }  (If)  ;  prism  of  the  second 
order,  oo  P2,  {  2110  }  (u),  and  prism  of  the  third  order, 
rcoPfl    ^3120^.   three  pvramids  of  the  first 

order,  fP,  }10l2}  (r),P,{1011}  (x),  and  2P,_  {  2021  }  (y)  ; 
one  pyramid  of  the  second  order,  2P2,  {  1121  }  (s)  ;  and 


one  of  the  third  order,  - 


,  ^{3121}  (m). 


As  other  examples  of  this  same  method  of  crystal- 
lization may  be  mentioned  the  three  similarly  consti- 
tuted compounds,  pyromorphite,  Pb5Cl(PO4)3  ;  mimete- 
site,  Pb5Cl(AsO4)3  ;  and  vanadinite,  Pb5Cl(VO4)3. 


122  CRYSTALLOGRAPHY. 

Rhombohedral  Hemihedrism.  The  selection  of  hex- 
agonal planes  by  alternate  dodecants  can  produce 
only  two  geometrically  new  forms,  since  on  only  two 
holohedrons  do  the  faces  belong  exclusively  to  single 
dodecants. 

The  extension  of  the  dihexagonal  planes  occupying 

alternate  dodecants  yields  a 
new  hemihedron,  bounded 
by  twelve  similar  scalene 
triangles,  and  called  the 
hexagonal  scalenohedron.  The 
two  corresponding  half- 
forms  possess  the  three  in- 
termediate planes  of  sym- 
metry, and  are  consequently 
congruent.  They  have  two 
sorts  of  polar  edges,  alter- 
nately more  acute  and  ob- 

Fio.  192.  FIQ.  193.  ,  .,      ,,       ,  ,       , 

tuse  ;  while  the  basal  edges 

form  a  zigzag  around  each  figure  (Figs.  192  and  193). 
These  two  forms  are  distinguished  as  positive  and 
negative,  and  their  symbols  are  written  : 

,  mPn          (7-77)  *.                        mPn 
+  -<p,      K{M&\;*       and g- ,       * 

A  scalenohedron  whose  polar  edges  are  equal  is  crys- 
tallographically  impossible,  for  the  reason  already 
given  for  the  dihexagonal  pyramid  (p.  110). 

*  The  Greek  letter  K  (xvlzVoS,-  inclined)  is  retained  to  designate 
the  indices  of  this  hemihedrism  in  order  to  emphasize  its  analogy  to 
the  sphenoidal  hemihedrism  of  the  tetragonal  system,  although,  as 
may  be  seen  by  an  inspection  of  the  figures,  the  new  forms  resulting 
in  this  case  are  not  inclined-face,  but  parallel-face  forms  (cf.  p.  42). 


THE  HEXAGONAL  SYSTEM.  123 

Each  face  of  the  hexagonal  pyramid  of  the  first 
order  occupies  one  dodecant,  and  this  form  is  there- 
fore capable  of  producing  rhombohedral  hemihedrons. 
The  extension  of  its  two  sets  of  alternating  planes 
yields  two  new,  congruent  half -forms,  which  are  each 


FIG.  194.  Fm.  195. 

bounded  by  six  similar  rhombs  and  called  rhombohe- 
drons  of  the  first  order  (Figs.  194  and  195).  Their 
symmetry  is  the  same  as  that  of  the  scalenohedrons  ; 
and,  like  these,  they  become  coincident  by  a  revolu- 
tion of  either  through  60°  about  its  vertical  axis. 
Their  symbols  are : 

and      -~, 

A 

Ehombohedrons  may  have  their  polar  edges  more 
or  less  acute  than  their  lateral  edges.  The  first  are 
known  as  acute,  and  the  latter  as  obtuse  rhombohedrons. 
The  limiting  form  between  these  two  kinds  of  rhom- 
bohedrons would  have  all  of  its  edges  similar,  and 
would  correspond  to  a  hexagonal  pyramid  whose  axial 
ratio  is  1  :  1.2247.  The  interfacial  angles  on  such  a 
form  would  all  be  90°,  and  it  would  therefore  not  differ 
geometrically  from  a  cube. 


124 


CRYSTALLOGRAPHY. 


The  four  following  figures  (196-199)  will  make  it 
evident  that  no  other  new  types  of 
forms  can  result  by  the  rhombohe- 
dral  selection  of  planes.  Each  of 
these  holohedrons  is  shaded  to  cor- 
respond to  a  disappearance  of  the 
faces  of  alternate  dodecants,  and  yet 
a  portion  of  each  holohedral  plane 
will  survive  in  every  case,  sufficient  to 
FIG.  196.  reproduce  the  form  as  it  was  before. 


Fio.  197. 


Fio.  198. 


Fio.  199. 


Abbreviated  Symbols  of  Rhombohedrons  and  Scaleno- 
hedrons.  .Ehombohedrons  and  scalenohedrons  are  of 
such  frequent  occurrence  on  natural  crystals  that  cer- 
tain abbreviated  symbols  have  been  suggested  for 
them  by  Naumann,  which  have  come  into  general  use. 

Rhombohedrons  are  designated  by  the  capital  initial 
J?,  preceded  by  the  vertical  parameter  of  the  hexagonal 
pyramid  from  which  they  are  derived.  Only. negative 
rhombohedrons  are  distinguished  by  a  sign ;  symbols 
without  signs  are  to  be  considered  as  positive.  For 
instance,  E  is  the  positive  rhombohedron  derived 


THE  HEXAGONAL  SYSTEM. 


125 


from  the  ground-form  or  fundamental  pyramid,  P. 


It  is  equivalent  to  the  symbol  -[-  - . 

A 


The  rhombo- 


hedron  —  872  is  the  negative  form  derived  from  the 
pyramid  of  the  first  order,  8  P  .  ^R  is  the  positive 
rhombohedron  derived  from  ^P,  etc. 

The  shortened  symbols  of  scalenohedrons  are 
formed  upon  a  different  principle.  In  every  scaleno- 
hedron  it  is  possible  to  inscribe  such  a 
rhombohedron  that  the  lateral  edges  of 
both  forms  shall  exactly  coincide  (Fig. 
200).  This  is  known  as  the  "rhombohe- 
dron of  the  middle  edges."  A  whole  series 
of  more  or  less  acute  scalenohedrons 
may  have  the  same  lateral  edges,  and 
hence  may  all  be  considered  as  derived 
from  the  same  "rhombohedron  of  the 
middle  edges"  by  increasing  its  vertical 
axis  by  different  rational  increments,  and 
then  joining  the  extremities  of  this  axis  to  the  lateral 
angles  of  the  rhombohedron.  Naumann's  abbreviated 
symbol  for  a  scalenohedron  consists  of  the  symbol  of 
its  "rhombohedron  of  the  middle  edges,"  m'R,  fol- 
lowed by  an  index,  n  ',  to  indicate  by  what  quantity  its 
semi-vertical  axis  is  multiplied.  The  new  symbol  is 
therefore  m'  'Rn  ',  in  which  m'  and  n'  are  quite  different 
quantities  from  the  two  parameters,  m  and  n,  of  the 
dihexagonal  pyramid,  of  which  the  scalenohedron  is  a 


FIG.  200. 


half-form    ± 

»         A 

In  order  to  be  able  to  transform  the  abbreviated 
into  the  full  symbols,  and  vice  versa,  we  must  know 


126  CRYSTALLOGRAPHY. 

the  relation  existing  between  these  four  quantities: 
ra',  ri  and  ra,  n. 

The  parameter  symbol  of  the  "  rhombohedron  of 
middle  edges"  belonging  to  a  scalenohedron  derived 

n      •          .  m(2  —  ri) 

from  na.  :  a2  :  -  -a3  :  me  is  a,  :  a  „:  oo  a.  :  —  --  '-c  • 
n  —  1  n 

hence  the  symbol  m'R,  expressed  in  terms   of  the 
parameters  of  its   corresponding  scalenohedron,  be- 


comes        -^  --  .     Any  particular  one  of  the  series 

of  scalenohedrons  having  this  same  "rhombohedron' 
of  the  middle  edges"  is  designated  by  the  index  n', 
whose  value  depends  upon  ra,  the  vertical  parameter 

ra> 
of  its  corresponding  dihexagonal  pyramid,  n'  =  —  ,  . 

To  obtain  the  abbreviated  symbol  for  a  scalenohe- 
dron from  its  full  symbol,  ±  —  ~~~  ,  we  have  : 

ra(2  —  n)  ,      m 

—  i  -  -  =  m'     and      —  =  ri  ; 
n  m 

and  hence,  for  the  reverse  transformation  : 
m'ri  =  m      and       =    ,     ,  —  n. 

The  indices  of  the  "  rhombohedron  of  the  middle 
edges"  belonging  to  a  scalenohedron,  K\hild  \  ,  are  2k—  h, 
0,  —  2k  -f-  h,  1.*  Hence,  if  the  indices  of  the  dihex- 

*  For  proof  of  this  see  Groth's  Physikalische  Krystallographie, 
3d  ed.,  p.  342. 


THE  HEXAGONAL  SYSTEM.  127 

agonal  pyramid  are  given,  we  have,  for  obtaining  the 
abbreviated  symbol  of  Naumann, 

-p-  =  m'      and      ^—j  =  n' ; 
or,  for  the  reverse  transformation, 


Example.  Given  the  dihexagonal  pyramid,  4P|  {  4131 
to  find  the  shortened  symbol  for  its  scalenohedron. 


and       2  =  w, 


A  — —      +*      •--"•       tlV  C4JJL.J.VL  x-| 

is  the  symbol  required.    Using  the  indices,  we  ob- 
tain the  same : 

?T-A=  2  =  ro'    and     ^—A  =  2  =  ri.     2B\* 
1  6  —  4 

Rhombohedral  Forms  in  Combination.  Rhombohedral 
combinations  differ  from  holohedral  hex- 
agonal combinations  only  when  rhombo- 
hedrons  or  scalenohedrons  are  present. 
These  forms  are,  however,  so  common 
that  their  more  frequent  combinations 
deserve  mention. 

A  rhombohedron  has  its  polar  angles 
blunted  by  the  faces  of  another  rhom- 
bohedron of  the  same  sign,  the  combina- 
tion edges  of  the  two  being  parallel  (Fig.  201). 

*  Dana  still  further  shortens  Naumann's  symbols  by  omitting  the 
initial  R  except  in  the  case  of  the  fundamental  rhombohedron. 
Thus  —  %  R  becomes  —  £  ;  4#,  4 ;  R*,  I3 ;  —  %R*,  —  ^  ;  etc.  On 
Miller's  method  of  designating  rhombohedral  forms,  see  Groth's 
Physikalische  Krystallographie,  3d  ed.,  p.  438. 


128 


CRYSTALLOGRAPHY. 


Much  more  frequent  are  combinations  of  rhombo- 
hedrons  of  opposite  signs.  These  generally  show 
truncations  of  the  polar  edges 
of  the  more  acute  form,  which 
can  only  be  accomplished  by  a 
rhombohedron  of  the  opposite 
sign  and  half  the  vertical  param- 
eter. For  instance,  R  truncates 
the  polar  edges  of  —  272  (r),  and 
is  in  turn  truncated  by  the 
faces  of  -  ±R(n)9  etc.  (Fig.  202). 
Figs.  203  and  204  show  combinations  of  rhombohedrons 
of  opposite  signs  where  the  above  relation  does  not 


FIG.  202. 


FIG.  203. 


FIG.  204. 


obtain.    In  the  first  case  the  forms  have  equal  vertical 
parameters,  R  and  — R ;  in  the  second,  a  rhombohedron 


FIG.  205. 


FIG.  206. 


THE  HEXAGONAL  SYSTEM. 


129 


FIG.  207. 


FIG.  208. 


is  modified  by  a  form  with  less  than  half  of  its  vertical 
parameter,  E  and  —  \R.  Figs.  205  and  206  show  a 
rhombohedroii  in  combination  with  a  prism  of  the  first 
order;  Figs.  207  and 
208,  the  same  form 
united  to  a  prism  of 
the  second  order. 

Combinations  of 
rhombohedronsand 
scalenohedrons  are 
very  manifold.  A 
scalenohedron  has 
its  polar  angles  re- 
placed by  its  "  rhombohedron  of  the  middle  edges"  so 
that  the  combination  edges  are  parallel  to  the  lateral 

edges  of  the  scaleno- 
hedron (Fig.  209).  In 
other  cases  either  the 
obtuse  or  acute  polar 
edges  of  the  scaleno- 
hedron are  truncated 
by  the  planes  of  a 
rhombohedron  (Fig. 
210);  or  the  polar  edges 
FIG.  209.  FIG.  210.  of  a  rhombohedron  are 

bevelled  by  the  faces  of  a  scalenohedron  (Fig.  211). 
Figs.  212  and  213  show  the  scalenohedron  in  combi- 
nation with  the  prisms  of  the  first  and  second  orders 
respectively. 

The  rhombohedral  hemihedrism  is  of  such  very  com- 
mon occurrence  that  only  a  few  of  its  most  prominent 
representatives  need  be  mentioned  here.  Examples 
of  somewhat  complex  combinations  of  rhombohedral 


130 


CRYSTALLOGRAPHY. 


FIG.  211. 


FIG.  212. 


FIG.  213. 


forms  are  given  in  the  two  following  figures.     Fig. 
214  represents  a  crystal  of  hematite  (Fe2O3)  bounded 


FIG.  214. 


FIG.  215. 


by  the  forms  R,  *jl011|(r);  §R,  *{3036J(tt);  i#, 
*{10l4K«);  IP2»  K  |2243!(ra);  and  \E\  *{4265}(i>  Fig. 
215  shows  a  crystal  of  calcium  carbonate  (calcite) 
with  the  forms  E,  /c{1011|(_p);  fR,  Ar{5052J(s);  4.R, 


;  J.R3,  /c  j  3124  J(0-  This  mineral  exhibits 
a  greater  variety  of  forms  than  any  other  rhombohe- 
dral  substance. 

Among  other  examples  of  this  mode  of  crystalliza- 


THE  HEXAGONAL  SYSTEM.  131 

tion  may  be  mentioned  the  elements  arsenic,  anti- 
mony, bismuth,  and  tellurium  ;  ice ;  corundum,  AlaO3 
(generally  with  forms  of  the  second  order,  and  there- 
fore often  apparently  holohedral) ;  magnesium  hy- 
droxide (brucite) ;  sodium  nitrate  (NaNO3)  and  the 
hydrous  silicate,  chabazite. 

TETARTOHEDRAL  DIVISION  OF  THE  HEXAGONAL  SYSTEM. 
Kinds  of  Tetartohedrism.  Tetartohedrisrn,  like  hemi- 
hedrism,  attains  its  maximum  importance  in  the  hex- 
agonal system.  We  may  develop  the  possible  kinds 
of  hexagonal  tetartohedrism  by  examining  the  effect 
of  a  simultaneous  application  of  two  different  kinds  of 
hemihedrism  to  the  planes  of  the  most  general  holo- 


FIG.  216.  FIG.  217.  FIG.  218. 

hedral  form,  as  was  done  in  the  preceding  systems. 
To  accomplish  this  we  may  imagine  each  of  the  three 
adjoining  figures  (216,  217  and  218)  laid  upon  one  of  the 
others  and  note  the  result.  A  union  of  the  third  with 
either  the  first  or  second  produces  a  survival  of  three 
planes  above  and  three  others  below  which  satisfy  the 
conditions  of  tetartohedrism  ;  while  the  second  super- 
posed upon  the  first  leaves  six  planes  on  the  lower 
half  of  the  crystal  and  none  on  the  upper  half. 

In  order  to  bring  out  this  result  still  more  clearly, 
we  may  number  the  planes  of  the  dihexagonal  pyra- 


132  CRYSTALLOGRAPHY. 

mid  as  is  indicated  in  Fig.  219,  and  then  write  out  the 
numbers  representing  the  twenty-four  planes  in  two 
rows,  as  follows  : 

(above)      123456789  10    11  12 
(below)      12'34'56'78*910'1112* 
If  now  we  erase  by  a  mark  to  the  right  those  planes 
which  would   disappear   by  one   kind   of   hexagonal 
hemihedrism,  and  by  a  mark  to  the  left  those  planes 
which  would  disappear  by  another  kind  of  hemihe- 
drism, we  can  obtain  the  three  following  results  : 

1.  We  may  combine  the  trapezohedral  and  rhombo- 
hedral  hemihedrisms  as  follows  : 

(above)     1-2    X  *  #  6    X&  ?  10     X  ^ 
(below)     XX'sV'fcX  7  j&^X'  11  V*' 

There  remain  (above)      ..  2  ...  6  ...  10  ... 
(below)      ....  3  ...  7  ...  11  . 

This  selection  yields  a  possible  tetartohedrism  which 

is  called  the  trapezohedral. 

2.  We  may  combine  in  the  same  manner  the  pyrami- 
dal and  rhombohedral  hemidrisms  as  follows : 

(above)     X*   X^  X  6    XK   #10    K^ 
(below)     ^V^'XVrs'XiQ'X  12' 

There  remain  (above)     ..  2  ...  6  ...  10  ... 
(below)     ....  4  ...  8  ....  12 

giving  another  possible  tetartohedrism  which  is  called 

the  rhombohedral. 

3.  Finally,  we  may  combine  the  trapezohedral  and 
pyramidal  hemihedrisms  : 

(above)      X  2    X  4  X  6    *  8    X 10    Xl2 

(below)   Xjr'^^'i-'jr'-'Y-ar'Uitf'  H  ^ 

There  remain  (above)   ..  2  ...  4 ...  6  ...  8  ...  10 ...  12 
(below)   


THE  HEXAGONAL  SYSTEM. 


133 


This  selection  does  not  satisfy  the  conditions  of  tetar- 
tohedrism,  but  produces  a  hemimorphism  in  the  direc- 
tion of  a  vertical  axis. 

Trapezohedral  Tetartohedrism,  The  simultaneous  ap- 
plication of  the  trapezohedral  and  rhombohedral  hemi- 
hedral  selections  to  the  faces  of  the  dihexagonal  pyra- 
mid produces  the  effect  shown  in  Fig.  219.  The 


FIG.  219. 


FIG.  220. 


extension  of  the   six  surviving   (white)  planes  gives, 
an  asymmetric  solid,  bounded  by  six   similar  trape- 
ziums.     The    same   result   is   secured   by   selecting 
one-half  of  the  faces  of  the  hexagonal  trapezohedron 
by  the  rhombohedral ;  or 
one  half  of  the  faces  of 
the  scalenohedron  by  the 
trapezohedral     method 
(Fig.  220).   The  four  trape- 
zohedral quarter-forms  of 
the  dihexagonal  pyramid 
form      two      enantiomor- 
phous  pairs  (Figs.  221  and 

222),  the  members  of  each          FIG.  221.  FIG.  222. 

pair   being  themselves  congruent.     They  are   called 
trigonal  trapezoedrons,  and  their  symbols  are  written : 


134 


CRYSTALLOGRAPHY. 


Positive  right-handed, 


mPn  .T 

—j-r,    or      --g— r,     Kr{kiU\ 


raj?71 


Negative  right-handed, 
mPn 

—  ~~A~r>       °r  ~~ 

Positive  left-handed, 
raPn, 

+-rl>  or   -T-( 

Negative  left-handed, 

raPw,  m^71. 


r,     *rr{tt5j 


congru- 
ent pair. 


congru- 
ent pair. 


The  union  of  a  right-  and  left-handed  form  would 
produce  a  scalenohedron ;  the  union  of  two  right- 
handed  or  two  left-handed  forms,  a  hexagonal  trape- 
zohedron. 

The  survival  and  extension  of  the  corresponding 
portions  of  the  planes  of  the  dihex- 
agonal  prism  produces  a  new  pris- 
matic form,  bounded  by  six  planes, 
intersecting  in  edges  which  are  alter- 
nately more  obtuse  and  more  acute. 
(Fig.  223).  This  is  called  the  di- 
trigonal  prism.  Two  corresponding 
forms  of  this  kind  are  derivable  from 
each  dihexagonal  prism,  which  differ 
only  in  position,  and  whose  symbols  are 


FIG.  223. 


and 


THE  HEXAGONAL  SYSTEM. 


135 


The  hexagonal  pyramid  of  the  second  order,  when 
subjected  to  analogous  selection,  retains  a  portion  of 
its  three  alternating  upper 
planes,  as  well  as  a  portion 
of  those  planes  directly 
below  them  (Fig.  224). 
The  extension  of  these 
planes  produces  a  solid 
bounded  by  six  similar 
isosceles  triangles,  inter- 
secting in  horizontal  basal 
edges.  This  form  is  called  FlG- 224- 

the  trigonal  pyramid.  There  is  a  congruent  pair  of 
them  corresponding  to  the  alternating  sets  of  pyram- 
idal planes,  their  symbols  being : 


mP2 


KT  \  Jckhl : 


and 


mP27 

' v. 


KT 


The  hexagonal  prism  of  the  second  order  is  only  a 
special  case  of  the  pyramid  of  the 
second  order,  where  the  basal 
edges  have  become  180°.  It  must 
therefore  yield  two  new  prismatic 
forms,  analogous  to  those  last  de- 
scribed, by  the  survival  of  its  al- 
ternate planes  (Fig.  225).  These 
are  called  trigonal  prisms,  and 
are  designated  by  the  symbols 


FIG.  225. 


>cr{1120};        and 


The  same  method  of  selection,  applied  to  the  hexag- 
onal pyramid  of  the  first  order,  yields  a  positive  and  a 


136 


CRYSTALLOGRAPHY. 


negative  rhombohedron,  which  are  not  further  modi- 
fied in  form  by  becoming  tetartohedral  (Figs.  226 
and  227). 


FIG.  236. 


FIG.  227. 


The  hexagonal  prism  of  the  first  order  suffers  no 
geometrical  change  whatever  on  be- 
coming tetartohedral,  as  may  be 
seen  from  Fig.  228 ;  and  the  same  is 
true  of  the  only  remaining  holohedral 
form,  the  basal  pinacoid. 

Trapezohedral  tetartohedral  forms 
may  also  be  regarded  as  the  product 
of  hemimorphism  in  the  direction  of 
the  three  lateral  axes,  and,  like  other 
hemimorphic  crystals,  they  are  pyro- 
electric.  Their  asymmetric  molecular  structure,  and 
consequent  enantiomorphism,  requires  that  they  should 
exhibit  circular  polarization,  and  this  is,  in  fact,  the 
case.  The  most  prominent  example  of  this  mode  of 
crystallization  is  offered  by  silica,  SiO2  (quartz).  A 
not  unusual  combination  of  forms  on  crystals  of  this 
substance  is  given  in  Fig.  229.  It  shows  the  prism  of 


FIG.  228. 


THE  HEXAGONAL  SYSTEM. 


137 


FIQ.  229. 


the  first  order  (m)  unchanged  ;  positive  and  negative 
rhombohedrons  (r  and  rf)  apparently 
only  hemihedral ;  while  the  trigonal 
pyramid  (s)  and  the  trigonal  trapezo- 
hedron  (x)  occur  as  true  quarter- 
forms. 

Other  examples  of  this  crystalliza- 
tion are:  mercuric  sulphide,  HgS  (cin- 
nabar) ;  the  dithionates  of  potassium 
(K2S2O6),  of  calcium  (OaS,O.  +  4  aq), 
of  strontium  (SrS2O6  -j-  4  aq),  of  ba- 
rium (BaS2O6  +  4  aq),  and  of  lead 
(PbS2O6  +  4  aq) ;  sodium -periodate  (NaIO4  -f-  3  aq)  ; 
benzil  (C14H10O2) ;  and  matico-stearopten  (C10H]6O). 

Rhombohedral  Tetartohedrism.  A  combination  of  the 
pyramidal  and  rhombohedral  methods  of  hemihedral 
selection  (Figs.  217  and  218,  p.  131)  results  in  the  survi- 
val of  three  planes  in  the  upper  half  of  the  dihexagonal 
pyramid  and  of  three  other  planes  in  the  lower  half. 
These  are  so  distributed  as  to  pro- 
duce, by  their  intersections,  a  figure 
which  does  not  differ  from  a  hemi- 
hedral rhombohedron  except  in  its 
position  with  reference  to  the  crys- 
tallographic  axes  (Fig.  230).  Each 
lateral  axis  terminates  on  one  side  of 
each  of  the  faces,  and  the  form  is 
called  the  rhombohedron  of  the  third 
order.  The  four  quarter-forms  de- 
rivable from  each  dihexagonal  pyramid  are  designated 
by  the  following  parameter  and  index  symbols : 
mPn  r 


FIG.  230. 


138 


CRYSTALLOGRAPHY. 


+ 


mPn   r 
I' 

I 


4 

mPn 


4    "r 

mPn     I 


These  four  quarter-forms  possess  three  planes  of  sym- 
metry, like  other  rhombohedrons,  and  are  therefore 
congruent.  The  position  of  these  planes  of  symmetry 
is,  however,  different  from  that  of  any  of  the  holohe- 
dral  hexagonal  planes  of  symmetry  (see  Fig.  159). 
This  same  method  of  selection,  applied  to  the  hex- 
agonal pyramid  of  the  second  or- 
der, is  capable  of  producing  two 
congruent  rhombohedrons  which 
again  differ  from  hemihedral  rhom- 
bohedrons only  in  their  positions 
(Fig.  231).  These  are  called 
rhombohedrons  of  the  second  order, 
and  they  stand  with  reference  to 
the  intermediate  axes  (p.  105)  just 
as  the  hemihedral  rhombohedrons 
do  with  reference  to  the  lateral  axes  of  reference. 
They  may  therefore  be  made  to  coincide  with  the  lat- 
ter by  a  revolution  about  their  vertical  axes  of  30°. 
Their  symbols  are : 


r 
I9 


FIG.  231. 


/-      4     r 

i  HfP;L     f 
' 


THE  HEXAGONAL  SYSTEM 


139 


The  same  method  of  selection,  if  applied  to  the  faces 
of  the  hexagonal  pyramid  of  the  first  order,  produces 
the  same  effect  as  the  rhombohe- 
dral  hemihedrism,  i.e.  two  rhombo- 
hedrons  of  the  first  order  (Fig.  232). 
Their  symbols  may  be  written : 


mP    r 


mP    I 


TtK 


TtK 


The  relative  positions  of  the  three  orders  of  rhom- 
bohedrons  is  well  illustrated  in  the  adjoining  linear 
projection  (Fig.  233),  where  those  of  the  first  order  are 
drawn  in  heavy,  those  of  the  second  order  in  faint,  and 
those  of  the  third  order  in  dotted  lines. 


Fia.  233. 


The   other    hexagonal    holohedrons   yield   in   this 
tetartohedrjsm  no  new  forms,     The  dihexagonal  prism 


140 


CRYSTALLOGRAPHY. 


FIG.  234. 


produces  two  hexagonal  prisms  of  the  third  order,  just 
as  it  does  in  the  pyramidal  hemihedrism.  The  other 
two  hexagonal  prisms  retain  all  their  six  faces. 

Examples  of  rhombohedral  tetarto- 
hedrism  in  the  hexagonal  system  are 
not  rare.  Fig.  234  shows  a  crystal 
of  copper  silicate  (dioptase)  with  a 
rhombohedron  of  the  third  order, 
-27#,  7tK\\- 13-14.6(  («),  in  combina- 
tion with  the  prism  of  the  second  order, 
ooP2,  J2110J  (ra),  and  the  negative 
rhombohedron  -272,  |0221J  (r). 

Fig.  235  represents  a  more  com- 
plicated crystal  of  the  beryllium  silicate,  phenacite, 
which  is  mainly  terminated  by  a  rhombohedron  of  the' 

third  order,  -  £  JT,  jr*|l382)  (x).   With  this  form  are 

associated  the  prisms  of  the  first  and  the  second  order, 
ooP,  jlOlOJ  (m),  and  ooP2  11120}  (a) ;  another  rhom- 

O  T>3 

bohedron  of  the  third  order,— ~r ,  7r>c{2131{   (s),  and 

the  two  rhombohedrons  of  the  first 
order,  R,  /cjlOllJ  (r),  and  -- 
/c{0112}.  With  this  substance  is  iso- 
morphous  the  corresponding  zinc  salt 
(willernite).  Still  other  examples  are 
the  carbonates,  magnesite,  MgCO3, 
and  dolomite,  (Ca,Mg)CO3 ;  possibly 
ilmenite,  FeTiO3 ,  and  some  other  sub- 
stances. 

Hemimorphism.  Hemimorphism  in  the  direction  of 
the  vertical  axis  (p.  42)  is  particularly  common  in  the 
hexagonal  system.  It  may  be  considered  as  produced 


FIG.  235. 


THE  HEXAGONAL  SYSTEM. 


141 


by  a  combination  of  the  trapezohedral  and  pyramidal 
hemihedrisms  (p.  133) ;  and,  as  it  generally  occurs  on 
crystals  which  are  also  rhombohedral,  these  may  be 
viewed  as  exhibiting  simultaneously  all  three  hemihe- 
drisms of  the  hexagonal  system.  Examples  of  such 
crystallization  are  offered  by  the  sulphides  of  zinc, 
ZnS  (wurtzite),  and  cadmium.  CdS 
(greenockite);  the  sulpharseni<Je  and 
sulphantimonicp  of  silver,  Ag3AsS3 
(proustite,  ruby  silver)  and  Ag3SbS3 
(pyrargyrite) ;  the  silicate  tourma- 
line ;  and  certain  artificial  salts.  A 
combination  observed  on  tourmaline 
is  represented  in  Fig.  236.  Its 
forms  are  the  prisms  ooP,  { 1010  [  (</), 
which,  as  a  result  of  hemimorphism, 
appears  as  a  trigonal  prism,  and 
the  scalenohedron  Ra,  /c{2131}  («),  and  the  two  rhom- 
bohedrons  R,  /cjlOll}  (R)  and  —  2#,  /c{0221}  (r\  at 
the  antilogue  pole,  while  only  one  of  these  forms,  R, 
occurs  at  the  analogue  pole. 

Hemimorphism  in  the  direction  of  the  vertical  axis 
produces  the  same  result  on 
holohedral,  trapezohedral  and 
pyramidal  hemihedral  crystals. 
Fig.  237  shows  the  hemimorphic 
development  of  the  holohedral 
zinc  oxide,  ZnO  (zincite),  from 
Stirling  Hill,  N.  J.  The  etched 
figures  observed  on  the  apparent- 
ly holohedral  silicate,  nepheliue, 
indicate  a  similar  mode  of  crystal- 
lization, disguised  by  complicated  twinning. 


FIG.  236. 

ooP2,  {1120J  (a); 


CHAPTEE  VI. 


THE  ORTHORHOMBIC  SYSTEM.* 
HOLOHEDBAL  DIVISION. 

Third  Class  of  Crystal  Systems.  According  to  the 
classification  of  the  crystal  systems  given  on  pages  44 
and  45,  all  those  whose  complete  forms  possess  no 
principal  axis  or  plane  of  symmetry  form  the  third 
or  anisometric  class.  These  systems  are  three  in 
number,  and  they  are  united  by  a  variety  of  common 
features,  among  which  is  their  optically  biaxial  char- 
acter and  the  fact  that  their  planes  are  all  referred  to 
three  unequal  and  therefore  not  interchangeable  axes. 
The  names  of  these  three  systems  are  the  Ortho- 
rhombic,  the  Monoclinic,  and  the  Triclinic,  each  of 
which  forms  the  subject  of  one  of  the  three  following 
chapters. 

Symmetry  of  the  Orthorhombic  System.  The  highest 
grade  of  symmetry  possessed  by  any  system  of  the 

third  class  is  that  of  the  or- 
thorhombic  system,  whose 
holohedral  forms  have  three 
secondary  planes  of  sym- 
metry at  right  angles  to 
one  another  (Fig.  238). 
These  planes  therefore  di- 
vide space  into  eight  simi- 


FIG. 


*  Also  called  the  rhombic,  prismatic,  and  trimetric  system. 

142 


THE  ORTHORHOMBIC  SYSTEM.  143 

lar  octants,  like  the  axial  planes  of  the  isometric  or 
tetragonal  systems. 

Axes.  The  orthorhombic  axes  of  reference  have 
their  directions  fully  determined  by  the  symmetry  of 
the  system.  They  are  all  axes  of  symmetry  and  must 
therefore  be  perpendicular  to  one  another;  but,  be- 
cause they  lie  in  secondary  planes  of  symmetry,  no  two 
of  them  are  interchangeable,  and  hence  they  must  all 
three  be  of  unequal  length.  Since  there  is  no  princi- 
pal axis  of  symmetry,  any  one  of  the  three  directions 
may  be  made  the  vertical  axis,  and  there  is  in  reality 
a  great  difference  in  the  usage  of  different  authors  in 
this  respect,  even  in  regard  to  crystals  of  the  same 
substance.  It  is  customary  to  place  the  orthorhom- 
bic crystal  in  such  a  position  that  the  longer  of  its 
two  lateral  axes  will  run  from  right  to  left.  This  is 
therefore  called  the  macrodiagonal,  while  the  shorter 
lateral  axis,  which  runs  from  back  to  front,  is  known 
as  the  brcwhydiagonal.  The 
letters  representing  the  differ- 
ent axes  are  further  distin- 
guished by  signs  written  over 
them  ;  thus,  a  short  sign  over  ~~°  Ijl^ 
the  brachydiagonal  (a),  a  long 
sign  over  the  macrodiagonal 
(6),  and  a  perpendicular  mark 
over  the  vertical  axis  (c)  (Fig.  FIG.  239. 

239).  The  distribution  of  the  positive  and  negative 
extremities  of  the  axes  is  the  same  as  in  the  isometric 
and  tetragonal  systems. 

Fundamental  Form  and  Axial  Ratio.  No  single  ortho- 
rhombic  crystal  form  (p.  35)  can  be  bounded  by  more 
than  eight  planes ;  because,  if  none  of  the  three  axes  are 


—a 


144  CRYSTALLOGRAPHY. 

interchangeable,  no  permutations  of  the  general  pa- 
rameter symbol  are  possible,  and  hence  only  one  plane 
belongs  to  an  octant.  All  orthorhombic  pyramids  are 
therefore  alike  in  the  number  and  distribution  of  their 
planes,  which  are  always  similar  scalene  triangles.  It 
is  possible  to  choose  any  pyramid  which  occurs  on  an 
orthorhombic  crystal  as  the  ground-form,  but  it  is 
customary  to  select  for  this  purpose  the  most  frequent 
or  prominent  pyramid,  or  the  one  which  will  yield  the 
simplest  indices  for  the  other  planes. 

When  the  choice  of  some  particular  pyramid  as  a 
ground-form  has  been  made,  the  axial  ratio  of  the  sub- 
stance to  which  it  belongs  may  be  calculated  from 
it.  The  inequality  in  the  lengths  of  all  three  axes 
produces  a  double  ratio,  in  which  the  length  of  the 
macrodiagonal  (jb)  is  taken  as  the  unit.  The  nature 
of  this  axial  ratio  is  the  same  as  in  the  tetragonal  sys- 
tem, except  that  in  place  of  a  single  irrational  quotient, 

f* 

-  (p.  83),  we  now  have  two  such  quotients  to  deter- 

a  (* 

mine  for  each  orthorhombic  substance,  -=•  and  7  .    These 

b         b 

quotients  are  called  the  crystaUographic  constants  of 
the  orthorhombic  system.  They  fix  the  axial  ratio," 
d  :  b  :  c,  and  may  be  determined  from  the  angles  of 
the  pyramid  selected  as  the  fundamental  form  as 
follows  (Fig.  240)  :  In  the  spherical  triangle  abc, 


cos  a  =  —.  —  ~?  ;  tg  .  a  =  a. 
sin    Z 


cos 


THE  ORTHORHOMBIC  SYSTEM. 


145 


One  axial  ratio  forms  the  basis  of  each  orthorhombic 
crystal  series  (p.  92) ;  and  hence  one  such  series  be- 
longs to  each  orthorhombic  substance.  No  matter 

4 


FIG.  5J40. 

what  pyramid  is  selected  as  the  fundamental  form,  all 
the  intercepts  on  the  brachydiagonal  axis  of  planes 
belonging  to  the  same  series  must  be  rational  mul- 
tiples of  its  quotient,  ^ ;  and  all  intercepts  of  planes 

on  the  vertical  axis  must  be  rational  multiples  of  its 

c 
quotient,  7-- 

Derivation  of  the  Holohedral  Orthorhombic  Forms. 
Since  the  most  general  orthorhombic  symbol,  nd:b:  me, 
is  capable  of  no  permutations,  it  can  only  represent 
an  eight-sided  pyramid,  essentially  like  the  funda- 
mental form  (Fig.  240).  This  formula  must  stand 
for  a  different  form  from  the  formula  a  :  nb  :  me,  since 
a  and  b  are  not  interchangeable.  It  is  customary 
always  to  make  the  lesser  of  the  two  lateral  parame- 
ters unity,  so  that  the  limiting  values  for  n  are  one 


146 


CRYSTALLOGRAPHY. 


and  infinity,  and  for  m  zero  and  infinity,  as  in  the 
tetragonal  system.  Since,  however,  the  parameter  n 
may  refer  to  either  of  the  two  dissimilar  lateral  axes, 
a  considerable  variety  of  form-types  may  result, 
which  can  best  be  considered  under  the  three  groups 
of  pyramids,  prisms,  and  pinacoids,  defined  on  p.  36. 

Pyramids.  These  are  forms  whose  planes  inter- 
sect all  three  axes.  All  orthorhombic  pyramids  are 
bounded  by  eight  similar  scalene  triangles,  which 
meet  in  three  kinds  of  edges  and  in  three  kinds  of 

solid  angles  (Fig.  240). 
It  is,  however,  usual  to 
distinguish  three  sorts 
of  such  pyramids,  ac- 
cording to  the  lateral 
axis  to  which  the  pa- 
rameter n  refers. 

There  is  .one  series 
or  zone*  of  pyramids 
both  of  whose  lateral 
parameters  are  equal 
to  unity,  a  :  b  :  me.  To 
this  series  the  funda- 
mental form  belongs, 
FlG.  24i.  and  it  is  therefore  called 

the  zone  of  unit  pyramids.  Its  limiting  forms  are  the 
basal  pinacoid,  where  m  =  0,  and  the  unit  prism  where 
m  —  oo  (Fig.  241).  The  general  parameter  and  index 
symbols  of  these  pyramids  are  mP,  \hJd\. 

On  one  side  of  the  zone  of  unit  pyramids  lie  those 
whose  lateral  parameter,  n  (>  1),  refers  to  the  macro- 


*  For  the  full  explanation  of  this  term,  see  Appendix,  p.  217. 


THE  ORTHORHOMBIC  SYSTEM. 


147 


diagonal  axis,  5.  These  are  called  macropyramids,  and 
their  general  symbols  are  written  a  :  nb  :  me,  mPn, 
{hkl}  (h>k)  (Fig.  242). 


Fia.  242. 

On  the  opposite  side  of  the  unit  pyramids  lies  a  third 
class  of  these  forms 
whose  lateral  parame- 
ter, n,  refers  to  the 
brachydiagonal  axis, 
a.  These  are  called 
br achy pyramids,  and 
their  general  symbols 
are  written  :  nd  :  b  :  me, 
mPn,{hkl\  (fc<fc)(Fig. 

243).  FIG.  243. 

For  every  possible  value  of  the  lateral  parameter,  n, 
there  is  a  vertical  zone  of  macro-  or  brachypyramids 
which  is  limited  by  the  basal  pinacoid  and  a  prism, 
as  is  the  zone  of  unit  pyramids. 

Prisms.  These  embrace  all  forms  whose  planes  are 
parallel  to  one  axis,  and  which  must  therefore  always 
have  one  sign  of  infinity  in  their  parameter  symbol,  or 
one  zero  in  their  index  symbol.  All  orthorhombic 


148 


CRYSTALLOGRAPHY. 


prisms  are  open  forms  (p.  36),  bounded  by  four  similar 
planes  meeting  in  two  sorts  of  edges,  but  without 
solid  angles.  We  may  again  distinguish  three  types 
of  such  prisms,  according  to  which  of  the  three  axes 
the  planes  are  parallel.  Although  they  all  belong 
equally  to  the  prismatic  type  of  forms,  it  is  customary 
to  call  those  whose  planes  are  parallel  to  either  of  the 
lateral  axes  domes. 

The  prisms  proper,  whose  planes  are  parallel  to  the 

vertical  axis,  e,  are 
limiting  forms,  in  one 
direction,  of  the  pyra- 
mids, and  must  there- 
fore be  of  three  kinds, 
like  these.  The  unit 
prism,  whose  lateral 
parameters  are  both 
Fl°-  344-  unity,  and  whose  sym- 

bols are  a  :  b  :  ooc,  ooP,  jllOJ  ;  the  macroprisms, 
whose  lateral  parameter  n  (>1)  refers  to  the  macro- 
diagonal  axis,  and  whose  symbols  are  a  :  nb  :  oo  c, 
cc  Pn,  \hkQ\  (h>k)-,  and  the  brachy prisms,  whose  lat- 
eral parameter  n  (>1)  refers  to  the  brachy  diagonal 
axis,  and  whose  symbols  are  nd  :  b  :  ooc,  oo  Pn,  \hkO\ 
(h<k)  (Fig.  244). 

The  other  two  types  of  prismatic  forms  or  domes 
are  distinguished  as  macrodomes  when  their  planes  are 
parallel  to  the  macrodiagonal  axis,  b  (Fig  245) ;  or  as 
brachydomes  when  their  planes  are  parallel  to  the 
brachydiagonal  axis,  a  (Fig.  246).  There  is  a  vertical 
zone  of  each  of  these  kinds  of  domes  whose  parameters 
ra  vary  between  zero  and  infinity.  The  general  sym- 
bols of  the  macrodomes  are  a  :  oo  b  :  me,  raPob,  \hQl\  ; 
and .  those  of  the  brachydomes,  oo  a  :  b  :  me, 


n^ 
i 

== 

*=• 

^=» 

i 

- 

1 

>=:tr 

, 

"V  ••^L.-.-ssJ 

ESS 

:j 

"*****«^^ 

THE  ORTHORHOMBIC  SYSTEM.  149 

\0kl\.  The  macro-  and  brachydomes  together  corre- 
spond in  their  positions  to  the  tetragonal  pyramids  of 
the  second  order  (p.  88). 


UJ— . 


FIG.  245.  Fio.  246. 

Pinacoids.  These  embrace  the  forms  whose  planes 
are  simultaneously  parallel  to  two  axes,  and  whose 
parameter  symbols  must  therefore  contain  two  signs 
of  infinity.  Each  of  these  is  an  open  form  bounded 
by  but  two  planes  which  are  parallel.  The  pinacoids 
of  themselves  therefore  have  neither  edges  nor  solid 
angles.  There  are  three  kinds  of  orthorhombic  pina- 
coids, as  well  as  three  kinds  of  pyramids  and  prisms. 
That  one  which  is  parallel  to  the  vertical  axis  and 
macrodiagonal  axis  is  called  the  wacropinacoid.  Its 
symbols  are  d  :  oo  b  :  oo  c,  oo  Poo,  { 100  j .  The  form  whose 
planes  are  parallel  to  the  vertical  and  brachydiagonal 
axis  is  called  the  brachypinacoid,  and  its  symbols  are 
GO  d  :  b  :  oo  c,  oo  Pob,  j  010  j .  Finally,  the  planes  parallel 
to  both  lateral  axes  are  called  basal  pinacoids,  and  their 
symbols  are  oo  a  :  oo  b  :  c  =  d  :  b  :  Oc  (p.  29),  OP,  [001 }. 
In  their  positions  the  orthorhombic  pinacoids  corre- 
spond to  the  faces  of  the  isometric  cube  (p.  56) ;  but 
the  six  planes  must  break  up  into  three  separate 
forms  so  soon  as  the  axes  cease  to  be  equivalent  and 
interchangeable. 

The  relations  of  limiting  forms  among  the  possible 


150 


CR  YSTALLOGRAPHY. 


types  of  orthorhombic  holohedrons  are  exhibited  in 
the  following  diagram  : 

Brachydiago 
Zone. 

OP 

I 


ra 


trachypyran 

OP 

1 

Hrf.S.        j 

Fundamental        Macropyramids.      Macrodiagonal 
Zone.                                                              Zone. 

OP             OP               OP 

1                 1                 1 

m 

IP 

m 

-  - 

m 

-  - 

IP* 

m 

Pn 

-      - 

P 

-  - 

Pn 

-  - 

Poo 

mPn 

-      - 

mP 

-  - 

mPn 

-  - 

wPob 

I                   I                  I                   I  I 

ooPdb ooPrc ooP ooPfl oo  Poo 


Prismatic  Zone. 


Orthorhombic  Forms  in  Combination.  As  the  grade  of 
symmetry  decreases  from  system  to  system,  the  num- 
ber of  planes  in  any  crystal  form  also  decreases ;  but 
at  the  same  time  the  number  of  forms  which  occur  in 
combination  upon  a  single  crystal  is  proportionately 
increased.  If,  however,  the  nature  of  the  separate 
types  of  forms  is  clearly  understood,  there  is  no  diffi- 
culty in  deciphering  these  more  complex  combina- 
tions, both  because  of  the  greater  simplicity  of  the 
forms  themselves,  and  also  on  account  of  their  zonal 
relation  to  each  other. 

The  number  of  substances  crystallizing  in  the  ortho- 
rhombic  system  is  very  great.  Of  these  we  shall 
mention  but  a  few,  to  serve  as  types  of  the  more  fre- 
quently occurring  combinations. 

The  three  orthorhombic  pinacoids  in  combination 
would  produce  a  figure  identical  with  the  cube  in 
shape,  and  yet  the  crystallographic  difference  between 
the  three  pairs  of  planes  may  manifest  itself  by  some 
physical  peculiarity,  as  in  the  case  of  the  three  un- 


THE  OETHORHOMBIC  SYSTEM. 


151 


equal   pinacoidal    cleavages    of    anhydrite    (calcium 
sulphate). 

The  fundamental  prism,  oo  P,  J110},  is  a  common 
form  on  orthorhombic  crystals.  It  may  be  combined 
with  two  pinacoids,  as  in  Fig.  247,  or  with  all  three. 
Fig.  248  shows  the  fundamental  prism  (p)  on  a  crystal 


FIG.  247. 
(Olivine.) 


FIG.  248. 
(Topaz.) 


of  topaz,  in  combination  with  another  prism,  oo  P2, 
1  120}  O');  the  basal  pinacoid,  OP,  {001}  (c)  ;  the 
brachydome,  Poo  {011}  (g)  ;  and  the  four  pyramids, 
P,  j  111  |  (o)  ;  fP,  |112!  (o');  tP,  1113}  (o");  and  |P2, 
{123}  (*)._ 

It  not  infrequently  happens  that  the  obtuse  pris- 
matic angle  in  the  orthorhombic  system  approaches 
very  nearly    to    120°.      Such    a 
prism,  combined  with  the  brachy- 
pinacoid,  would  simulate  a  hex- 
agonal  prism.     If  these  forms  are 
terminated  by  fundamental  pyra- 
mids and  brachydomes,  the  whole 
combination  often  closely  resem- 
bles a  hexagonal  crystal.     Fig.  249  shows  such  a  com- 


FIG.  249. 
(Chalcocite.) 


152 


CRTSTALLOGEAPHT. 


bination  observed  on  the  copper  sulphide,  chalcocite, 

with  the  forms:  OP,  J001J  (c)  ;  oo  Poo,  {010}  (6); 

ooP,  {110|  (p);  2Pob,  {021}  (?);  |P&,  1023}  (?');  P, 

{111}  (o);  andfP,  {113}(<>;). 

The  orthorhombic  combinations  are  far  too  numer- 
ous to  be  described  systemati- 
cally. A  number  are,  however, 
appended,  with  the  symbols  of 
their  planes,  in  order  to  familiar- 
ize the  student  with  some  of  the 
more  complex  types. 

Fig.  250  (sulphur)  shows   the 
forms:  P,{111}(^);  iP,jll3}«; 
OP,  S001|  (c)  ;  and  Pec,  {011}  (n). 
Fig.  251  (silver  sulphide,  acanthite)  shows  a  combina- 

tion of  the  following  forms:  oo  Poo,  {010}  (a);  oo  Poo, 

1  100  }  (ft)  ;  OP,  1  001  }  (c)  ;  oo  P,  j  110  }  (I)  ; 

oo  P2,  {120}  (m);  Pfc,  {010}  (o)  ; 

3P&,  {031}  (e);  £Pob,  {102}  (d);  P2, 


FIG.  250. 


JP,  J113}  (r);  and  2P4,  J142J  (n).    If, 

however,  as  is  still  more  common,  the 

form  ra  be   assumed   as   the   funda- 

mental prism,  instead  of  /,  the  axis 

running  from  back  to  front  becomes 

twice  as  long  and  must  then  be  re- 

garded as  the  macrodiagonal  instead 

of  the  brachydiagonal.     In  this  case 

the  above  symbols  for  the  planes  of  the  figure  be- 

come :  oo  Poo,  j  100  }  (a)  ;  oo  Poo,  j  010  }  (b)  ;  OP,  {  001  }  (c)  ; 

oo  P2,  {120}  (Q;  oo  P,  {110}  (m);  Poo,  {101}  (o)  ;  3Pob, 

{301}  (6);  Pob,{011}(d);  P,{111}(&);  2P2,  {121}  (p)  ; 

3P3,  {131}  (s);  |P2,  {123}  (r);  and  2P2,  {211}  (n). 


FIG.  251. 
(Acanthite.) 


THE  ORTHORHOMBIC  SYSTEM. 


153 


Fig.  252  reproduces  a  combination  of  orthorhombic 
forms  observed  on  crystals  of  aluminium  orthosilicate, 
andalusite(A!2SiO6) :  ooPoo,  {010}  (a);  ooPob,  {100}  (6); 
OP,  {001}  (c);  oo  P,  {110}  (m);  ooP2,  {210}  (Z);  ooP2, 
{120}  (n);  Pob,  {101}  (r);  P,  |lll|(p);  P*,  {011}  (s); 
2P2,  {121}  (k). 


m 


m 


FIG.  252. 
(Andalusite.) 


m 


m 


FIG.  253. 
(Cerussite.) 


Fig.  253  shows  a  crystal  of  lead  carbonate  (cerus- 
site)  which  is  bounded  by  the  following  forms :  OP, 
|001}(c);  oo  Pob,  {010}  (a);  oo  P,  { 110 }  (m)  ;  P,  { 111 }  (p) ; 
fP,  {112}  (o);  iPob,  {102}  (*/);  ^Pao,  {012}  (a?);  Pc5b, 
{011}  (k);  2P2,  {121}  (5);  2P2,  {211}  (w). 

Fig.  254  exhibits  a  combination  occurring  on  the  fer- 
ro-magnesian  metasilicate,  hypersthene  [(Fe,Mg)SiOJ. 
ooPo),{010}(a);  oo  Pob,  { 100 }  (b) ;  oo  P,  { 110 }  (m) ;  ooP2, 
{120}  (n) ;  iP&,  {014}  (A);  P,  {111}  (o);  P2,  {212}  (c); 
2P2,  {241}  (i)i  |P|,  {232}  (M). 

Fig.  255  represents  a  crystal  of  the  corresponding 
ferro-magnesian  orthosilicate,  olivine  [(Fe,Mg)SiOJ  : 
oo  Pob,  J010}  (a);  OP,  |001}  (c);  ooP,  |110J  (n);  ooP2, 
J120}(«);  ooP§,S130;(r);  Pob,  j  101  i  (c«) ;  Poo,  { 011}  (A); 
2P&,  J021}  (k);  4Pob,  {041}  (t) ;  P,  Jill}  (e);  JP, 
J112}  (o);  2PS,  {121}  (/);  3P§,  {131}  (I). 


154 


CRYSTALLOGRAPHY. 


Orthorhombic  crystals  are  often  disproportionately 
elongated  in  the  direction  of  one  axis,  when  they  are 


m 


FIG.  254. 
(Hypersthene.) 


called  prismatic,  columnar,  or  acicular  in  their  habit. 
In  such  cases  the  axis  of  elongation  is  generally  se- 
lected as  the  vertical  axis  (see  Fig.  251).  In  other 
cases  the  crystals  are  disproportionately  shortened  in 
the  direction  of  one  axis,  when  their  habit  is  said  to 
be  tabular  or  lamellar  (p.  14). 


HEMIHEDRISM  IN  THE  ORTHORHOMBIC  SYSTEM. 

Kinds  of  Hemihedrism.  As  the  number  of  planes  be- 
longing to  a  single  crystal  form  decreases  with  the 
symmetry  of  the  system,  the  possibility  of  the  exist- 
ence of  partial  forms  likewise  decreases.  Thus  hemi- 
hedrism  is  of  minor  importance  in  the  systems  without 
any  principal  plane  of  symmetry,  but  it  nevertheless 
exists. 

It  is  possible  to  imagine  one  half  of  the  eight  planes 
which  bound  an  orthorhombic  pyramid  to  be  chosen 
in  three  different  ways. 


THE  ORTHORHOMBIC  SYSTEM.  155 

1.  We  may  select  the  planes  alternately.     Tliis  pro- 
duces, as  we  shall  presently  see,  the  disappearance  of 
all  the  planes  of  symmetry,  but  the  surviving  planes 
satisfy  the  fundamental  conditions  of  hemihedrism. 

2.  We   may   select   the   planes   by  alternate  pairs. 
This  results  in  the  disappearance  of  all  but  one  of 
the   planes  of  symmetry,  while  the  resultant  forms 
differ  in  no  respect  from  those  of  the  succeeding  or 
monoclinic  system. 

3.  We  may  select  the  planes  in  groups  of  four  about 
the  extremities   of   either  axis.     This  results  in  the 
disappearance  of  only  one  of  the  three  planes  of  sym- 
metry, and  in  the  production  of  forms  which  are  not 
hemihedral,  but  hemimorphic. 

Sphenoidal  Hemihedrism,  There  is  therefore  but  a 
single  kind  of  true  hemihedrism  possible  in  the 
orthorhombic  system,  and  this  is  caused  by  the  sur- 
vival of  alternate  planes  on  the  most  general  form,  the 
orthorhombic  pyramid.  The  extension  of  these  planes 
until  they  intersect  must  result  in  the  production  of 
an  asymmetric  figure,  bounded  by  four  similar  scalene 
triangles,  meeting  in  three  sorts  of  edges.  Such  a 
form  is  exactly  analogous  to  the  isometric  tetrahedron 
or  tetragonal  sphenoid  (p.  101),  and  is  called  the  ortho- 
rhombic  sphenoid.  The  two  sphenoids  derivable  in  this 


FIG.  256.  FIG.  257. 

way  from  every  orthorhombic  pyramid  are,  on  account 
of  their  lack  of  all  planes  of  symmetry,  enantiomor- 


156 


CRYSTALLOGRAPHY. 


phous    forms  (p.  66)    (Figs.   256    and   257). 
general  parameter  and  index  symbols  are : 


Their 


K  \JiE\\ 


and 


mPn 


I,      *{W\ 


It  is  evident  that  this  method  of  selection  can  pro- 
duce no  new  forms  from  orthorhombic  prisms  or 
domes,  since  each  plane  of  these  forms  corresponds  to 
two  contiguous  pyramidal  planes.  Still  less  can  it 
give  rise  to  new  forms  when  applied  to  the  pinacoids, 
whose  planes  correspond  to  four  pyramidal  faces. 
Orthorhombic  substances  which  exhibit  sphenoidal 
hemihedrism  are  not  altogether  uncom- 
mon, especially  among  organic  salts.  As 
examples  of  this  method  of  crystalliza- 
tion may  be  mentioned  the  silicate,  leuco- 
phane  ;  the  hydrous  sulphates  of  magne- 
sium (Epsom  salts),  MgSO4  +  7  aq  (Fig. 
258),  and  zinc  (zinc  vitriol),  ZnSO4  -f-  7  aq ; 
acid  potassium  tartrate  (cream  of  tar- 
tar); potassium  sodium  tartrate  (Eo- 
chelle  salts) ;  tartar  emetic ;  lactose  ;  and 
mycose. 

Sulphur  crystals  also  rarely  exhibit  a  sphenoidal  de- 
velopment of  their  planes. 

Hemimorphism.  Hemimorphism  in  the  direction  of 
one  of  the  three  orthorhombic  axes  is  also  not  uncom- 
mon. In  such  cases  it  is  usual  to  select  the  hemimor- 
phic  axis  as  the  vertical  axis.  The  two  planes  of 
symmetry  which  intersect  in  this  axis  remain  in  hemi- 
morphic  crystals  as  such,  while  the  plane  normal  to 
it  ceases  to  be  a  plane  of  symmetry. 


FIG.  258. 


THE  ORTHORHOMBIC  SYSTEM. 


157 


As  examples  of  orthorhombic  hemimorphism  may 
be  cited  ammonium-magnesium  phosphate  (struvite), 
NH4MgPO4  +  6  aq  (Fig.  259),  which  shows  above  (anti- 
logue  pole)  the  faces  :  Poo,  {101}  (r) ;  P3b,  J011  j  (q)  • 
4  Poo,  { 041 }  (q') ;  while  below  (analogue  pole)  are  the 
forms  OP,  { 001  { (c) ;  JPob,  { 103 }  (/);  and  oo  Poo,  { 010'}  (6). 


FIG.  259. 


Fia.  260. 


Other  hemimorphic  substances  are  basic  zinc  silicate 
(calamine,  hemimorphite),  Zna(OH)2SiO3  (Fig.  260)  ; 
resorcine,C6H6O2;  triphenylmethan,(C6H5)CH;  lactose, 
C12H22On.  The  last-named  substance  is  both  hemi- 
hedral  and  hemimorphic. 


CHAPTEK  VII. 


THE  MONOCLINIC  SYSTEM.* 

Symmetry  and  Axes.  No  complete  crystal  form,  i.e. 
one  bounded  by  pairs  of  parallel  planes  (p.  18),  can 
possess  two  planes  of  symmetry,  without  at  the  same 
time  possessing  three.  The  grade  of  symmetry  next 
lower  than  the  orthorhombic,  which  can  possibly 
characterize  a  crystal  system,  must  therefore  be  pro- 
duced by  a  single  plane  of  symmetry. 

The  existence  of  such  a  single  plane  of  symmetry 
determines  the  direction 
normal  to  it  as  an  axis  of 
symmetry,  and  this  is  the 
only  direction  which  is  fixed 
for  the  whole  system 
(Fig.  261).  The  other  two 
necessary  axes  of  reference 
must  lie  in  the  plane  of  sym- 
metry, bat  their  directions 
in  this  plane  are  a  matter  of 
arbitrary  choice,  to  be  de-  FlG.  26i. 

cided  in  the  case  of  each  substance  as  shall  be  most 
convenient.  Both  of  these  other  axes  must  therefore, 
in  all  cases,  be  perpendicular  to  the  axis  of  symmetry, 
but  they  may  make  any  angle  whatever  with  each 
other. 

It  is  customary  to  place  crystals  having  only  one 
plane  of  symmetry  in  such  a  position  that  this  plane 

*  Known  also  as  the  monosymmetric,  clinorhomMc  or  oblique  system. 

158 


THE  MONOCLINIC  SYSTEM.  159 

shall  stand  vertical,  and  the  axis  normal  to  it  shall 
run  from  right  to  left.  Furthermore,  one  of  the  two 
directions  which  are  chosen  as  axes  in  the  plane  of 
symmetry  is  made  vertical,  and  the  other  is  so  placed 
that  it  inclines  downward  toward  the  front  (Fig.  261). 
In  this  way  we  see  that  the  three  axes  directly  de- 
ducible  from  the  single  plane  of  symmetry  are  two 
directions  at  right  angles,  and  a  third  which  is  at  right 
angles  to  the  first  but  inclined  at  any  angle  to  the 
second.  Hence  crystal  forms  whose  planes  are  refer- 
able to  such  axes  are  called  monoclinic. 

The  axis  of  symmetry,  which  is  always  made  to  run 
from  right  to  left,  is  designated  by  the  letter  b  sur- 
mounted by  a  straight  line,  — ,  and  called  the  ortho- 
diagonal  axis.  The  axis  which  is  made  vertical  is 
represented  by  the  letter  c,  as  in  the  orthorhombic 
system,  and  called  the  vertical  axis,  while  the  oblique 
axis  is  designated  by  a  surmounted  by  an  oblique 
sign,  \  and  called  the  clinodiagonal  axis.  This  position 
differs  from  that  in  the  orthorhombic  system  in  so  far 
as  the  axis  b  may  or  may  not  be  longer  than  d. 

Fundamental  Form  and  Crystallographic  Constants.  Just 
as  in  the  orthorhombic  system,  so  in 
the  monoclinic,  any  pyramid  occur- 
ring on  a  crystal  may  be  assumed  as 
aground-form,  from  whose  angles  the 
axial  ratio  for  the  particular  substance 
may  be  calculated.  This  ground-form 
here  determines  not  merely  the  rela- 
tive unit  lengths  of  the  three  axes 
d  :  b  (=1):  c,  but  also  the  inclination  of  FlQ-  262- 

the  two  axes  d  and  c  to  each  other  (Fig.  262).  This  angle, 
which  is  designated  by  the  Greek  letter  (3,  is  a  crystal- 


160 


CRYSTALLOGRAPHY. 


w 


lographic  constant  for  each  monoclinic  crystal  series, 

Ct  (*    4£- 

just  as  are  the  two  irrational  quotients   j-    and    — 

Although  any  monoclinic  pyramid  could  possibly 
serve  as  a  fundamental  form,  the  choice  of  this  is  not 
altogether  arbitrary,  since  it  is  very  desirable  that  the 
axes  d  and  c  should  be  parallel  to  prominent  planes, 
which  then  become  pinacoids  or  prisms.  One  of  them, 
at  least,  is  usually  determined  by  the  habit  of  the 
crystal,  and  is  made  to  coincide  with  a  direction  of 
elongation  or  cleavage.  If  different  authors  select  dif- 
ferent ground-forms  for  crystals  of  the  same  substance, 
the  cry  stall  ographic  constants  derived  from  these 
must  always  bear  a  rational  relation  to  one  another. 
Derivation  of  the  Holohedral  Monoclinic  Forms,  The 
most  general  monoclinic  symbol, 
nd  :  b  :  me,  stands  for  only  half  as 
many  planes  as  are  represented  by 
the  corresponding  orthorhombic 
symbol  (p.  145).  As  was  explained 
in  Chapter  II  (p.  35),  the  presence 
of  any  plane,  A,  making  any  angle 
with  a  single  plane  of  symmetry, 
WXYZj  necessitates  the  presence  of 
the  similar  plane  B ;  while  to  both 
FIG.  sea.  A  and  B  there  must  be  a  parallel 

plane  A  and  Bf  (Fig.  263).  Four  is  therefore  the 
greatest  number  of  planes  that  can  belong  to  any 
monoclinic  form. 

*  The  process  of  calculating  the  crystallographic  constants  from 
the  angles  of  the  fundamental  form  becomes,  in  the  monoclmic  and 
triclinic  systems,  too  complex  to  be  given  here,  and  must  be  sought 
in  works  like  those  of  Dana,  Klein  or  Liebisch,  where  mathematical 
crystallography  is  treated  of  at  length. 


THE  MONOCLINIC  SYSTEM. 


161 


FIG.  264. 


The  four  planes  composing  any  monoclinic  pyramid 
belong  either  altogether  to 
the  acute  or  altogether  to 
the  obtuse  octants.  Such 
forms  cannot  of  themselves 
enclose  space ;  but  if  their 
planes  are  extended,  they 
form  open  prisms  (Fig.  264) ; 
and  in  fact  they  would  really 
be  monoclinic  prisms  if 
either  of  the  axes  lying  in 
the  plane  of  symmetry  had 
been  selected  so  as  to  run 
parallel  to  their  intersec- 
tion edges. 

Monoclinic  pyramids,  because  they  occupy  but  half 
of  the  eight  octants,  are  called  hemi-pyramids,  and  are 
furthermore  distinguished  as  positive  and  negative, 
according  as  their  planes  belong  to  the  acute  or  obtuse 
octants.*  The  parameter  and  index  symbols  of  cor- 
responding positive  and  negative  hemi-pyramids  are 
-f-  mPn,  \  hid  \  and  —  mPn,  { Jikl } .  Two  such  forms 
together  correspond  to  the  planes  of  the  orthorhom- 
bic  pyramid,  but  they  are  nevertheless  completely 
independent  and  holohedral.  Their  planes  are  of  dif- 
ferent shapes,  and  either  alone  entirely  satisfies  the 
conditions  of  monoclinic  symmetry. 

The  possible  monoclinic  holohedrons,  derived  from 
the  most  general  symbol  by  giving  limiting  values  to 

*  This  universally  accepted  nomenclature  is  unfortunate,  since  the 
four-faced  monoclinic  pyramids  are  in  no  sense  hemihedral.  The 
upper  positive  form  likewise  cuts  the  negative  end  of  the  clino- 
diagonal  axis,  but  this  usage  was  adopted  by  Natimaun  so  as  to  make 
the  cos  ft  (when<  90°)  a  positive  quantity,  since  this  is  an  important 
factor  in  the  calculation  of  monoclinic  forms. 


162  CRYSTALLOGRAPHY. 

one  or  both  of  its  variable  parameters,  agree  exactly 
with  those  in  the  orthorhombic  system,  except  that 
all  forms  whose  planes  belong  exclusively  to  acute  or 
obtuse  octants  become  positive  and  negative  hemi- 
forms.  All  forms,  on  the  other  hand,  whose  planes 
lie  simultaneously  in  acute  and  obtuse  octants  are 
identical  in  the  two  systems. 

Pyramids.  There  are  three  sorts  of  positive  and 
negative  monoclinic  hemi-pyramids  corresponding  to 
the  three  sorts  of  orthorhombic  pyramids ;  and,  like 
them,  named  from  the  lateral  axis  to  which  their  larger 
parameter,  n,  refers.  These  are  : 

1.  The  zone  or   series   of   unit  pyramids,  both  of 
whose   lateral  parameters  are   unity.     Their  general 
symbols   are  -f-  mP,    \7ihl\    and  —  mP,    \hJd\.     They 
extend,  according  to  the  value  of  the  vertical  parame- 
ter, m,  between  the  unit  prism  where  m  becomes  GO 
and  the  basal  pinacoid  where  it  becomes  0. 

2.  On   one  side   of    the   unit   pyramids    lie   those 
whose  larger  lateral  parameter,  n,  refers  to  the  ortho- 
diagonal  axis.     These  are  called  hemi-orthopyramids, 
and  their  general  symbols  are  -|-  mPn,  \  hkl  \  (h>  k)  and 
—  mPn,  \hkl\  (h>k). 

3.  On  the  opposite  side  of  the  unit  pyramids  are 
such   as  have  their  larger  lateral  parameter  n  refer- 
ring   to    the   clino-diagonal   axis.     These   are   called 
hemi-dinopyramids,  and   their   symbols   are   -f-  mPn, 
\hE\  (h<k)  and  -mPh,  \hE\  (h<k). 

Prisms.  There  are  three  sorts  of  prismatic  types 
whose  planes  are  parallel  to  one  of  the  three  axes  of 
reference.  As  in  the  orthorhombic  system,  those 
which  are  parallel  to  the  vertical  axis  are  called  prisms 
proper,  and  the  others  domes. 


THE  MONOCLINIC  SYSTEM.  163 

The  three  kinds  of  vertical  prisms  correspond  to  the 
three  kinds  of  monoclinic  pyramids,  viz.:  unit  prisms, 
whose  symbols  are  d  :  b  :  oo  c,  oo  Pt  {HOf ;  orthoprisms, 
whose  symbols  are  d:nb:ccc,  co  Pn,  {hkO}  (A>&); 
and  dinoprisms,  whose  symbols  are  nd  :  b  :  oo  c,  oo  Ph, 
[hJcO]  (h<k).  The  planes  of  all  of  these  forms  belong 
at  once  to  both  an  acute  and  an  obtuse  octant  and 
hence  do  not  in  any  way  differ  from  the  four-sided 
orthorhombic  prisms. 

The  monoclinic  domes  are  named  from  the  axes  to 
which  their  planes  are  parallel,  orthodomes,  whose  sym- 
bols are  d  :  GO  b  :  me,  mPob,  { hQl } ;  and  dinodomes,  whose 
symbols  are  oo  d  :  b  :  me,  mPob  {OJd}.  The  orthodomes 
have  their  planes  confined  entirely  to  two  acute  or  to 
two  obtuse  octants,  and  hence  consist  of  two  indepen- 
dent forms  of  two  planes  each,  -f-  mPab,  {hOl}  and 
—  raPob  \  Wl\,  called  positive  and  negative  hemi-ortho- 
dorneSi  The  clinodomes,  on  the  other  hand,  have  their 
planes  lying  at  once  in  an  acute  and  an  obtuse  octant, 
and  cannot  therefore  break  up  into  hemi-forms. 

Pinacoids.  These,  as  in  the  other  systems,  consist 
of  pairs  of  planes  which  are  parallel  to  two  of  the  axes. 
Their  planes  must  belong  simultaneously  to  two  acute 
and  two  obtuse  octants,  so  that  they  cannot  differ  from 
the  corresponding  orthorhombic  forms  (p.  149).  They 
are  known  as  the  orthopinacoid,  parallel  to  the  vertical 
and  orthodiagonal  axis  d  :  oo  b  :  oo  6,  oo  Poo,  J100};  the 
clinopinacoid,  parallel  to  the  vertical  and  clinodiagonal 
axis  oo  d  :  b  :  oo  c,  oo  Poo  ,  j 010 } ;  and  the  basal  pinacoid, 
parallel  to  the  axis  of  symmetry  (the  orthodiagonal) 
and  that  direction  which  has  been  assumed  as  t^e 
clinodiagonal  axis,  oo  d  :  oo  5  :  c,  or  d  :  b  :  Oc,  OP,  {001}. 

The  relations  of  limiting  forms  in  the  monoclinic 
system  are  shown  in  the  following  diagram : 


164 


CRYSTALLOGRAPHY. 


Clinodiago- 
nal  Zone. 

OP 

1 

Clino- 
pyramids. 

OP 

1 

F 

\mdamental 
Zone. 

OP 

1 

Ortho- 
pyramids. 

OP 

1 

Orthodiago- 
nal  Zone. 

OP 

1 

m 

- 

m 

m 

- 

4^ 

±v* 

Po> 

- 

±    Ph 

- 

±     P 

- 

±     Pn 

- 

±     Poo 

mPoo 

" 

±mPh 

- 

±mP 

- 

±  mPn 

- 

±  mPoo 

I 
oo  Poo  -  -  oo  Ph  - 


I  I 

.  .  co  P oo  Pn 

Prismatic  Zone. 


-  -  -  oo  Poo  - 


Monoclinic  Forms  in  Combination.  Monoclinic  combi- 
nations are  more  varied,  and  at  the  same  time  more 
common,  than  those  of  any  other  system.  It  is,  how- 
ever, impossible  to  convey  a  satisfactory  idea  of  their 
proportions  by  means  of  plane  figures  ;  and  hence  the 
use  of  models  and  actual  crystals  is  more  necessary 
than  ever  to  render  them  completely  intelligible. 
The  first  step  in  deciphering  the  combinations  pre- 
sented by  such  models  or  crystals  is  to  bring  them 
into  correct  position,  i.e.  to  seek  out  their  plane  of 
symmetry  and  to  place  it  vertically  and  at  the  same 
time  to  direct  it  toward  the  observer.  The  orthodiag- 
onal  axis  will  then  run  horizontally  and  from  right 
to  left.  Thus  far  the  position  is  imperative  for  all 
monoclinic  crystals,  but  the  selection  of  the  remaining 
axes  is  a  matter  of  convenience  in  each  particular  case. 
Of  all  the  planes  which  are  perpendicular  to  the  plane 
of  symmetry  (that  is,  which  belong  to  the  orthodiag- 
onal  zone)  any  parallel  pair  may  be  chosen  as  the 
orthopinacoids,  and  any  other  pair  as  the  basal  pin- 
acoids.  This  choice  will  of  course  be  regulated  by 
the  habit  of  the  crystal,  and,  in  the  case  of  all  common 
substances,  it  is  already  established  by  a  usage  which 


THE  MONOCLINIC  SYSTEM.  165 

cannot  be  ignored.  Whichever  planes  are  made  the 
orthopinacoids  will  determine  the  direction  of  the 
vertical  axis,  while  those  selected  as  the  basal  pinacoids 
must  be  placed  so  as  to  slope  downward  toward  the 
observer,  thus  conditioning  the  direction  of  the  clino- 
diagonal  axis.  When  the  crystal  has  in  this  manner 
been  brought  completely  into  position,  it  is  possible 
to  designate  its  other  planes.  All  those  lying  in  the 
orthodiagonal  zone,  except  the  pinacoids,  become 
positive  or  negative  hemi-orthodomes.  All  other 
planes  really  belong  to  one  class  of  four-sided  pris- 
matic forms,  and  which  of  them  become  prisms,  which 
clinodomes,  and  which  pyramids,  depends  entirely 
upon  the  choice  of  the  ortho-  and  basal  pinacoids. 

On  the  crystal  of  iron-vitriol  represented  in  Fig.  265 
the  only  plane  whose  value  is  absolutely  fixed  is  the 
plane  of  symmetry,  or  clinopinacoid 
(6).  It  is  customary  to  make  c  the 
basal  pinacoid,  and  p  the  fundamen- 
tal prism,  whence  q  becomes  a  clino- 
dome,  o  a  negative  hemi-pyramid,  and 
r'  and  r  plus  and  minus  hemi-ortho- 
domes. We  might,  however,  turn  the 
crystal  so  as  to  make  c  the  orthopin- 

•  J  J.T-  •  J  V          J  FlG-  265' 

acoid,  q  the  prism,  and  p  a  clinodome  ; 
or  we   might  even  make   r'  the  basal  pinacoid,  and 
r  the  orthopinacoid,  when  o  would  become  the  prism, 
c  a  hemi-orthodome,  and  p  and  q  both  pyramids. 

The  following  examples  of  monoclinic  combinations 
observed  on  natural  crystals  will  be  found  useful. 

Fig.  266  shows  a  crystal  of  orthoclase  with  the  forms 
ooPoo,  jOlOK^f);  OP,  {001 }  (P);  oo  P,  jllOj  (T) ; 
+  Poo,  {101}  (x) ;  +2P6b,  {201}  (y) ;  +  P,  {111}  (o) ; 
2Poo,  {021}  (n). 


166 


CRT8TALLOGRAPHT. 


Fig.  267  represents  a  crystal  of  arsenic  disulphide 
(realgar),   As2S2,    with    the    forms    OP,    {001}    (P) ; 


M 


FIG.  266. 


FIG.  267. 


FIG.  2G8. 


ooPoo,  {010}  (r);    ooP,  {110}  (M)  ;     ooP2,  {210}  (Z) ; 
Po>,  {Oil |  (71);  and    +  P,  {111}  (a). 

Fig.  268  represents  a  crystal  of  calcium-magnesium 
metasilicate  (diopside),  (CaMg)SiO3,  the  symbols  of 
whose  forms  are:  OP,  {001}  (c);  oo  Poo ,  {010}  (6); 
oo Poo,  {100}  (a);  ooP,  {110}  (m);  +  2P,  [%}  (o); 
and  -P  {111}  0). 

Fig.  269  shows  a  crystal  of  calcium  silico-titanate 
(sphene),  CaTiSi05,  whose  forms  are  : 
OP,  {001  jfo);  oo  Poo  ,  {010}  (g);  o>  Poo, 
{100J  (c);  ooP,  {110}  (r);  PSo, 
\011\(e);  +P,\llll(t);  ^  P, 
{lllKn);  -2P,  J221}(V). 

Monoclinic  crystals  are  very  often 
elongated  parallel  to  some  direction 
in  the  plane  of  symmetry,  which 
direction  is  then  generally  made  the 
vertical  axis  (see  Fig.  266).  In  cases 
where  the  basal  pinacoid  is  fixed  by  some  physical 
property  like  cleavage,  the  elongation  is  in  the  direc- 
tion of  the  clinodiagonal  axis.  This  is  often  the  case 


FIG.  269. 


THE  MONOGLINIC  SYSTEM. 


167 


with  crystals  of  the  mineral  feldspar,  as  is  shown  in 
Fig.  270  which  represents  nearly 
the  same  combination  of  forms  as 
Fig.  266. 

In  still  other  cases  the  elonga- 
tion of  the  crystal  may  be  in  the 
direction  of  the  orthodiagonal  axis, 
whose  position  is  determined  by 
the  plane  of  symmetry.  A  crystal 
of  the  silicate  mineral,  epidote, 
shows  this  (Fig.  271).  Its  forms 
are:  OP,  {001}  (Jtf);  ooPob,  {100}  (T);  +  2Pob, 
{201}  (0;  +P6b,  {101}  (r); 

Monoclinic    crystals    may 

^___      also     be     flattened     in     the 

FIG  ^  direction    of    their    axis    of 

symmetry,  as  in  the  case  of 

sanidine  ;  or  in  a  direction  perpendicular  to  it,  as  in 
the  case  of  mica. 


FIG.  270. 


HEMIHEDRISM  IN  THE  MONOCLINIC  SYSTEM, 

Kinds  of  Hemihedrism.  It  is  possible  to  select  one 
half  of  the  four  planes  composing  a  monoclihic  hemi- 
pyramid  in  three  and  only  three  different  ways,  which 
are  illustrated  in  the  three  following  figures  (272,  273 
and  274). 

1.  We  may  select  the  two  parallel  planes  B  and  E' 
(Fig.  263)  belonging  to  two  diagonally  opposite  octants 
as  shown  in  Fig.  272,  which  would  produce  forms  in  no 
way  differing  from  those  in  the  following  or  triclinic 
system. 


168 


CRYSTALLOGRAPHY. 


2.  We  may  select  the  two  planes  A  and  B  at  one 
extremity  of  the  vertical  axis ;  or,  in  other  words,  the 


FIG.  272. 


Fio.  273. 


FIG.  274. 


two  which  intersect  in  the  plane  of  symmetry  (Fig.  273). 

By   such   a   selection   the  surviving   planes  are   still 

equally  distributed  about  both  extremities  of  the  axis 
of  symmetry,  and  they  therefore 
satisfy  all  the  conditions  of  a  true 
hemihedrism  (p.  41).  Fig.  275 
shows  such  a  hemihedral  crys- 
tal of  pyroxene,  whose  forms  are 
the  same  as  those  represented  in 
Fig.  268. 
FIG.  275.  3.  We  may  select  the  two  planes 

B  and  A',  which  intersect  at  one  end  of  the  axis  of  sym- 
metry (Fig.  274).    This  will  result  in 

the  production,  not  of  hemihedrism, 

but  of  hemimorphism  in  the  direction 

of   the   axis   of   symmetry.      Such  a 

development  is  illustrated  by  pentacid 

alcohol    (quercite),     C6H]2O6.       The 

forms  on  the  crystal  of  this  substance 

shown  in  Fig.  276  are :  OP,  { 001 }  (c) ; 

oo  P,    {110}    (j>);    +   *^S    {101}  (r); 

and  Poo,  J011J  (q).     The  latter  form  occurs  .only  on 


FIG.  276. 


THE  MONOCLINIC  SYSTEM. 


169 


one  side  of  the  plane  of  symmetry.  Other  examples 
of  monoclinic  hemimorphism  are  afforded  by  the 
organic  salts,  tartaric  acid  ;  cane-sugar  ;  cinchotinine 
nitrate;  cinchendibromide  ;  and  campheroxim. 

Forms  produced  by  the  second  of  these  three  modes 
of  selection  retain  their  plane  of  symmetry,  while  by 
both  the  other  methods  of  selection  this  disappears. 

Tetartohedrism  in  the  Monoclinic  System.  The  inde- 
pendent occurrence  of  one  quarter  of  the  planes  of  the 
most  general  monoclinic  form  would  result  in  the 
most  simple  form  conceivable, 
i.e.  a  single  plane.  This  would 
not  be  separable  from  hemihe- 
drism  in  the  triclinic  system, 
where  each  holohedron  consists 
of  two  parallel  planes.  The  re- 
peated occurrence  of  isolated 
pyramidal  planes  on  crystals  of  cane-sugar  has  led 
some  authors  to  consider  this  substance  as  possibly 
an  example  of  such  crystallization  (Fig.  277). 


\ 


Fia.  277. 


~ 


CHAPTEE  VIII. 

THE  TRICLINIC  SYSTEM.* 

Symmetry  and  Axes.  The  only  possible  grade  of 
symmetry  lower  than  that  possessed  by  the  mono- 
clinic  system  (p.  158)  consists  in  the  absence  of  any 
plane  of  symmetry  whatever.  We  have  already  en- 
countered in  all  the  preceding  systems  partial  forms 
which  were  altogether  without  symmetry ;  but  such 
forms  do  not  generally  possess  the  property,  which 
belongs  to  every  crystallographic  holohedron,  of  hav- 
ing their  planes  distributed  in  parallel  pairs  (p.  18). 
The  forms  of  the  present  system,  if  they  are  to  be  re- 
garded as  holohedral,  must  possess  this  property,  as 
they  do. 

Since  there  is  no  plane  of  symmetry,  the  presence 
of  any  face  necessitates  only  the  presence  of  its  oppo- 
site parallel  face  (see  Fig.  31,  p.  35) ;  so  that  all  forms 
of  the  triclinic  system  are  alike  in  consisting  of  only 
two  parallel  planes. 

The  selection  of  the  three  axes  of  reference  is  alto- 
gether a  matter  of  convenience,  since  none  of  them  can 
be  an  axis  of  symmetry.  Any  three  pairs  of  planes 
occurring  on  a  crystal  may  be  chosen  as  the  pinacoids, 
and  so  be  made  to  determine  the  positions  of  the  axes. 
The  axes  will  therefore  differ,  not  merely  in  their  unit 
length,  but  also  in  their  directions  for  all  triclinic  crys- 
tals of  different  substances  (crystal  series)  ;  while  their 

*  Also  called  the  asymmetric,  clinoi'homboidal  or  anorthic  system. 

170 


TEE  TRICLINIG  SYSTEM.  171 

directions  may  also  differ  for  crystals  of  the  same 
substance,  if  they  are  differently  chosen  by  different 
authors.  But  however  the  axes  are  selected,  they  must 
always  be  of  different  unit 
lengths,  and  must  also  be  ob- 
liquely inclined  to  each  other, 
whence  the  designation  of  this 
system  as  the  triclinic.  When 
their  directions  are  decided 
upon,  it  is  customary  to  place  -^' 

the  crystal  in  such  a  position 
that  one  axis  shall  stand  verti- 
cally (the  vertical  axis,  c)  (Fig. 
278) ;  while  the  longer  of  the 
two  remaining  axes  inclines 

downward  toward  the  right  (macrodiagonal,  5),  and  the 
shorter  downward  toward  the  front  (brachydiagonal,  a). 
The  three  oblique  interaxial  angles  are  designated  by 
Greek  letters  as  follows :  c/\b=a;  c/\d  =  ft', 
b  A  &  —  y.  When  in  the  above  position,  all  the  inter- 
axial  angles  in  the  upper  right  front  octant  will  be 
greater  than  90°. 

Fundamental  Form  and  Crystallographic  Constants,  In 
the  triclinic  system  there  are  five  constants  to  be  de- 
termined for  each  crystal  series.  These  are  the  three 
interaxial  angles,  #,  ft  and  y,  whose  values  are  given 
by  the  inclinations  of  the  faces  selected  as  pinacoids, 

a  f* 

and  the  two  irrational  quotients,  =r  and  j-t  which  de- 
termine the  axial  ratio,  a  :  b  :  c,  just  as  they  do  in  the 
orthorhombic  and  monoclinic  systems.  These  five 
constants  cannot  all  be  determined  from  a  single  form  ; 
nor  is  this  necessary,  since  a  triclinic  form  cannot 


172  CRYSTALLOGRAPHY. 

occur  except  in  combination  with  others.  If  the  direc- 
tions of  the  axes  are  known,  both  ratios  may  be  calcu- 
lated from  any  pyramidal  face  which  is  selected  as  the 
unit  pyramid ;  or  the  first  may  be  more  easily  obtained 
from  a  prism,  and  the  second  from  a  dome  face.  The 
macrodiagonal  axis,  b,  is  here,  as  in^the  preceding  sys- 
tems, assumed  as  the  unity  for  the  axial  ratio. 

Derivation  of  the  Holohedral  Triclinic  Forms.  When 
the  fundamental  forms  have  been  selected  for  any 
triclinic  substance,  and  its  crystallographic  constants 
thereby  determined,  the  designation  of  all  other  possible 
forms  agrees  closely  with  that  employed  in  the  ortho- 
rhombic  system.  We  have  but  to  remember  that  no 
triclinic  crystal  form  can  consist  of  more  than  two 
planes,  and  that  these  must  always  be  parallel,  as  ex- 
plained on  p.  35. 

Pyramids.  Any  plane  on  a  triclinic  crystal  which 
intersects  all  three  of  the  directions  assumed  as  the 
crystallographic  axes  is  a  pyramid.  Furthermore,  any 
pyramid  may  be  selected  as  the  fundamental  form,  as 
was  stated  above.  When  this  selection  has  been  made, 
there  will  be  a  possible  zone  of  unit  pyramids  whose 
lateral  parameters  are  the  same,  but  whose  vertical 
parameters  are  different  from  those  of  the  fundamental 
form. 

Pyramidal  planes  occurring  on  one  side  of  the  unit 
pyramids,  with  their  parameters  on  the  brachydiag- 
onal  axis  greater  than  unity,  are  called  brachypyra- 
mids  ;  while  those  occurring  on  the  other  side  of  the 
unit  pyramids,  with  their  parameters  on  the  macro- 
diagonal  axis  greater  than  unity,  are  called  macro- 
pyramids,  as  in  the  orthorhornbic  system  (p.  147). 

Since  the  entire  absence  of  symmetry  in  the  tri- 


THE  TRIGLINIC  SYSTEM.  173 

clinic  system  necessitates  but  two  parallel  planes  for 
any  complete  holohedral  form,  no  triclinic  pyramid 
can  occupy  more  than  two  of  the  eight  octants  into 
which  the  axial  planes  divide  space.  By  the  oblique 
intersections  of  these  axial  planes,  however,  these 
octants  are  not  all  similar,  as  in  the  orthorhombic 
system,  but  fall  into  four  pairs  of  dissimilar  octants, 
the  similar  members  of  each  pair  being  diagonally 
opposite.  The  two  parallel  pyramidal  j  lanes  occur- 
ring in  any  one  of  the  four  pairs  of  octants  are  entirely 
independent  of  any  planes  occurring  in  the  other 
octants,  even  when  these  have  the  same  parameters. 
Because  the  complete  triclinic  pyramid  can  occupy  but 
one  quarter  of  the  octants  it  is  called,  by  analogy  with 
the  monoclinic  hemi-pyramid  (p.  161),  a  tetra-pyramid 
(see  Fig.  31,  p.  35). 

The  symbols  for  the  triclinic  pyramids  are  like  those 
of  the  other  systems,  except  that  an  accent  is  written 
near  Naumann's  initial  P,  to  designate  which  one  of 
the  four  dissimilar  octants  on  the  front  of  the  crystal 
the  form  belongs  to.  Thus  the  four  triclinic  tetra- 
pyramids  whose  parameters  are  all  unity  are  written : 

P'jlllf  for  the  upper  right-hand  octant ; 
'P  \  111 }  for  the  upper  left-hand  octant ; 
P t  \  111  \  for  the  lower  right-hand  octant ; 
f  \  111  \  for  the  lower  left-hand  octant. 

Prisms.  All  forms  of  the  prismatic  type  (p.  36)  are 
composed  of  planes  which  are  parallel  to  one  of  the 
crystallographic  axes,  and  therefore  common  to  two  con- 
tiguous octants.  In  the  orthorhombic  system  these 
forms  are  all  composed  of  four  planes  which  correspond, 
in  the  triclinic  system,  to  two  independent  forms  of  two 


174  CRYSTALLOGRAPHY. 

planes  each.  By  analogy  with  the  orthorhombic 
designation  (p.  148),  these  are  called  hemi-prisms  when 
they  are  parallel  to  the  vertical  axis ;  hemi-brachy- 
domes  when  they  are  parallel  to  the  shorter  lateral 
axis ;  and  hemi-macrodome^  when  they  are  parallel  to 
the  longer  lateral  axis.  Like  the  pyramids,  their 
position  is  indicated  by  accents  written  in  the  symbols, 
though  two  of  these  are  now  necessary  to  show  the  two 
octants  to  which  their  planes  belong.  Thus  the 
symbols  for  the  prismatic  forms  two  of  whose  param- 
eters are  unity  become  in  the  triclinic  system 

oo  P '  \  110 }  right-hand  hemi-prism  ; 
'(coP  1 110}  left-hand  hemi-prism  ; 
ffo1  |011[  right-hand  upper  hemi-brachydome ; 
'Poo,  { Oil }  left-hand  upper  hemi-brachydome  ; 
'Poo7  1 101}  upper  front  hemi-macrodome  ; 
,-Pob,  {101}  lower  front  hemi-macrodome. 

Pinacoids.  The  pinacoidal  planes  are  parallel  to  two 
axes,  and  therefore  belong  equally  to  four  contiguous 
octants.  The  pinacoids  consist  of  but  two  parallel 
planes  in  the  orthorhombic  system  (p.  149),  and  there- 
fore cannot  be  different,  except  in  their  relative  incli- 
nations, in  the  triclinic  system.  Their  names  and 
symbols  are  identical  with  those  in  the  orthorhombic 
system,  viz.  : 

ooPob  j  010 }  parallel  to  a  and  c,  the  brachy-pinacoid ; 

ooPobjlOOJ  parallel  to  b  and  c,  the  macro-pinacoid ; 

OP  { 001 1  parallel  to  a  and  5,  the  basal  pinacoid. 

In  the  following  diagram  the  relation  of  the  differ- 
ent form-types  possible  in  a  triclinic  crystal  series  is 
represented  as  it  has  been  in  each  of  the  preceding 
systems. 


THE  TRICLINIC  SYSTEM. 


175 


Brachy-                                               F,,«^««,  />«/«;                                             Macro- 
diagonal          Brachy-pyramids.       fundamental       Macro-pyramids.        diagonal 
Zone                                                                                                                          Zone. 

OP               OP                OP                OP               OP 

1                     1                       1                       1                     1 

1 

'Ip»  ip*' 

'ip  V 

71     _     1 

IP*' 

,m 

m         m 

m       m 

m         m 

m 

'ip*; 

_ 

--P#    --Pnl 

_ 

ip  IP, 

_ 

1           1 
—  JRI    —  Pn. 

--Poo 

m 

,m         m 

,m       m 

jin         m 

ym 

I 

_ 

\            \ 

_ 

I          i 

_ 

\            i 

_ 

I 

,p<*>' 

'Pn       Pn' 

/p      p/ 

'Pn    Pn' 

^ob' 

'p*; 

- 

fn       Pn, 

- 

^  ^ 

- 

fn    Pnt 

- 

.Pob, 

i 

_ 

\             \ 

_ 

I          i 

— 

\            \ 

1 

,mP&' 

'mPn     mPn' 

'mP    mP' 

fmPn     mPn! 

'»p«' 

'mP*, 

- 

tmPn     mPhl 

- 

,mP    mP  t 

- 

tmPn    mPn/ 

- 

/mPoo/ 

1 

\            i 

1                   | 

\            I                I 

ooPob 

- 

/oaPKoaP*/ 

- 

',  OOP  OOP/ 

- 

/oo  P^l  ooPw/ 

- 

ooPob 

Prismatic  Zone. 

The  derivation  of  the  index  symbols,  corresponding 
to  these  parameter  symbols  of  Naumann,  is  in  all  re- 
spects the  same  as  in  the  preceding  systems,  and  can 
therefore  present  no  difficulty. 

Triclinic  Forms  in  Combination.  Since  the  number  of 
planes  belonging  to  each  form  is  in  the  triclinic  sys- 
tem the  least  possible,  the  variety  of  possible  combina- 
tions is  proportionately  greater.  Such  combinations 
are  not,  however,  difficult  to  decipher.  Familiarity 
with  them  can  only  be  acquired  by  study  of  figures, 
models,  or  crystals  of  triclinic  substances,  a  few 
examples  of  which  are  here  appended  by  way  of 
illustration. 

Fig.  279  shows  a  combination  observed  on  calcium 
hyposulphite  (CaS2O3  +  6  aq).  If  we  assume  the  face 


176 


CRYSTALLOGRAPHY. 


c  as  the  basal  pinacoid  OP  { 001 } ,  and  b  as  the  brachy- 
pinacoid,  GO  P&  {010},  then  the  symbols  of  the  other 
planes  become:  oo  P/,  {110}  (j>);  /ooP,  {110} 
and  ,P&',  {011} 


FIG.  279.  Fio.  280.  FIG.  281. 

Fig.  280  reproduces  a  crystal  of  hydrous  copper 
sulphate  whose  forms  are:  OP,  {001}  (c) ;  GO  Poo, 
{f&j(a);  oo  P&,  1 010}  (6);  oo  P/,  {110j(n);  >  P, 
{110 1  («);  >P2,  {120J  ©;  and  P',  {lllf  (*). 

Fig.  281  shows  a  crystal  of  the  manganese  meta- 
silicate  (rhodonite),  MnSiO3,  whose  forms  are:  OP, 
|001}  (c);  ooPco,  {100}  (6); 
{100}  (a);  oo  P/,  {110}  (n) ; 
{011}  (A);  'Poo',  {101}  (o);  and.Pob,, 
{101}  (a). 

Fig.  282  shows  a  crystal  of  the 
same  substance  as  the  last  with  a 
somewhat  different  combination  and 
placed  in  a  different  position.  The 
same  planes  have  the  same  letters  as  in 
Fig.  281,  but  they  now  receive  differ- 
ent symbols,  as  follows :  OP,  { 001 }  (a); 
oo  Poo,  {100}  (o);  oo  Poo,  {010)  (s); 
ooP/, {110}  (6);  >P,  jUOl(«);ftP> 
{221}  (n);  and 


THE  TRIGLINIC  SYSTEM. 


177 


FIG.  283. 


Fig.  283  shows  a  complicated  crystal  of  calcium 
aluminium  unisilicate  (anorthite),  CaAlaSi2O8,  whose 
planes  are  given  the  symbols: 
OP,  {001}  (P);  ooP3b,{010j  (M); 
ooP/,  {110}  (0;  >P,  {110KZ7); 
ooP3/,|130}(/);  >P3,{130}(*); 
P',{lll\(m);  'P,  {111}  (a);  P,, 
Jill}  (o);  yP,  {HI  |  (j>);  4P2,, 
{241}(i;);  'Poo',  {101}  (Q;  ,2Pob,, 
J201}  (?/);  ,Pdb',  {011}  (e);  ^Pob', 
{021J  (r);  and  'Pdb^  {011}  (n).  The  obliquity  of  the 
axes  in  this  crystal  is  so  slight  that  it  approaches 
closely  to  its  limiting  form  in  the  monoclinic  system, 
which  actually  occurs  on  the  corresponding  potassium 
feldspar,  orthoclase. 

How  crystals  may  find  their  limiting  forms  in  sys- 
tems of  a  higher  grade  of  symmetry  has  already  been 
explained  in  Chap.  IV  (p.  91).  To 
make  this  clearer,  the  student  will 
find  it  a  useful  practice  to  select 
some  form  of  a  high  grade  of  sym- 
metry and  consider  what  forms  in 
systems  of  lower  symmetry  it  limits. 
For  instance,  the  isometric  rhombic 
dodecahedron  (Fig.  284),  whose 
symbol  is  oo  a  :  a  :  a,  is  bounded  by 
twelve  planes,  which  correspond  to  two  tetragonal,  three 
orthorhombic,  four  monoclinic,  and  sixtriclinic  forms, 
as  is  shown  in  the  diagram  at  the  top  of  the  next  page. 
The  limiting  forms  in  different  systems  become  of 
practical  value  in  explaining  the  so-called  "  optical 
anomalies"  exhibited  by  many  crystals  whose  physical 
behavior  is  not  strictly  in  accord  with  what  seems  to 


FIG.  284. 


178  CRYSTALLOGRAPHY. 

Tetragonal.          Orthorhombic.         Monoclinic.  Triclinic. 


be  tlieir  external  form ;  and  in  this  connection  they 
will  be  again  referred  to  in  Chapter  IX. 

CLASSIFICATION  OF  ALL  CRYSTAL  PORKS  BY  THEIR 

SYMMETRY. 

The  following  list  of  all  the  holohedral,  hemihedral, 
tetartohedral  and  hemimorphic  divisions  of  each  sys- 
tem, classified  according  to  their  planes  of  symmetry, 
may  prove  of  use.  Roman  numerals  indicate  principal 
planes  of  symmetry,  and  Arabic  numerals  secondary 
planes. 

Nine  planes  of  symmetry. 

1.  Isometric,  holohedral.     Ill  +  6.     (Fig.  51.) 

Seven  planes  of  symmetry. 

2.  Hexagonal,  holohedral.     1+3  +  3.     (Fig.  164) 

Six  planes  of  symmetry. 

3.  Isometric,  tetrahedral  hemihedral.     6.     (Fig.  94.) 

4.  Hexagonal,  hemimorphic.      3  +  3.      (Angle  30°.) 

(Fig.  237.) 

Five  planes  of  symmetry. 

5.  Tetragonal,  holohedral.     1+2  +  2.     (Fig.  126.) 

Four  planes  of  symmetry. 

6.  Tetragonal,   hemimorphic.      2  +2.      (Angle   45°.) 

(Fig.  158.) 


THE  TRICLINIC  SYSTEM.  179 

Three  planes  of  symmetry. 

1.  Isometric,  parallel-face  hemihedral.      3.      (Angle 
90°.)    (Fig.  76.) 

8.  Hexagonal,  rhombohedral  hemihedral.    3.    (Angle 

60°.)    (Fig.  194.) 

9.  Orthorhombic,  holohedral.    3.    (Angle  90°.)    (Fig. 

240.) 

Two  planes  of  symmetry. 

10.  Tetragonal,    sphenoidal,  hemihedral.     2.     (Angle 

90°.)     (Fig.  155.) 

11.  Orthorhombic,   hemimorphic.      2.      (Angle    90°.) 

(Fig.  259.) 

One  plane  of  symmetry. 

12.  Tetragonal,  pyramidal  hemihedral.     I.    (Fig.  150.) 

13.  Hexagonal,  pyramidal  hemihedral.    I.    (Fig.  185.) 

14.  Monoclinic,  holohedral.     1.     (Fig.  265.) 

15.  Monoclinic,  hemihedral.     1.     (Fig.  275.) 

No  plane  of  symmetry. 

16.  Isometric,  gyroidal  hemihedral.     (Fig.  66.) 

17.  Isometric,  tetartohedral.     (Fig.  109.) 

18.  Tetragonal,  trapezohedral  hemihedral.     (Fig.  148.) 

19.  Tetragonal,  trapezohedral  tetartohedral. 

20.  Tetragonal,  sphenoidal  tetartohedral. 

21.  Hexagonal,  trapezohedral  hemihedral.     (Fig.  183.) 

22.  Hexagonal,    trapezohedral    tetartohedral.       (Fig. 

221.) 

23.  Hexagonal,    rhombohedral    tetartohedral.      (Fig. 

234.) 

24.  Orthorhombic,  sphenoidal  hemihedral.    (Fig.  256.) 

25.  Monoclinic,  hemimorphic.     (Fig.  276.) 

26.  Triclinic,  holohedral.     (Fig.  279.) 


CHAPTER  IX. 

CRYSTAL  AGGREGATES. 

Kinds  of  Aggregates.  In  the  preceding  chapters  we 
have  considered  only  the  crystal  individual  (p.  16)  as 
a  unit,  and  entirely  independent  of  its  relations  to 
other  individuals,  either  of  the  same  or  of  other  kinds. 
Crystals,  however,  often  conform  in  their  groupings  to 
certain  definite  laws  which  therefore  become  an  integral 
part  of  crystallography. 

The  ideal  crystal  individual  is  invariably  a  poly- 
hedron whose  mterfacial  angles  are  all  less  than  180°; 
the  presence  of  re-entering  angles  on  a  crystalline 
surface  therefore  indicates  the  union  of  two  or  more 
individual  crystals. 

In  a  crystal  aggregate  (p.  16)  the  individuals  may 
be  all  of  one  kind,  or  of  different  kinds.  In  the  first 
instance  the  molecular  arrangement  may  be  completely 
parallel  throughout  all  the  individuals,  which  are  then 
only  separated  from  each  other  by  their  external 
planes.  Such  aggregates  are  called  parallel  growths. 
In. other  cases  there  may  be  no  relation  whatever  be- 
tween the  molecular  orientation  of  two  contiguous  in- 
dividuals. Groups  of  crystals  formed  in  this  way  are 
called  irregular  aggregates. 

If  we  conceive  of  the  resultant  of  all  the  attractive 
and  repellent  forces  belonging  to  the  crystal  molecule 

180 


CRYSTAL  AGGREGATES.  181 

as  resolved  into  three  components  not  at  right  angles, 
we  can  understand  how  the  parallelism  between  the 
molecular  structures  of  two  individual  crystals  of  the 
same  substance  may  be  partial.  This  has  been  already 
explained  in  Chapter  I  (p.  8,  Figs.  5,  6,  and  7).  If  all 
three  axes  of  two  similar  molecules  are  parallel,  the 
orientation  is  identical,  as  in  parallel  growths ;  if  none 
of  the  axes  are  parallel,  the  orientation  is  wholly  differ- 
ent, as  in  irregular  aggregates ;  but  if  one  or  two  of 
the  three  axes  of  one  molecule  are  parallel  to  the  cor- 
responding axes  of  the  other  molecule,  the  orientation 
is  neither  complete  nor  wanting,  but  partial.  Such  a 
position  as  that  last  described  may  be  imagined  as 
produced  by  a  revolution  of  one  of  a  pair  of  completely 
parallel  molecules,  through  an  angle  of  180°  about 
either  of  its  axes.  Two  crystal  individuals  which  have 
this  kind  of  a  relative  position  are  said  to  be  tivins. 

In  case  the  crystal  aggregate  is  formed  of  individuals 
of  different  substances,  there  may  be  almost  complete 
parallelism  of  orientation  if  the  different  crystals 
possess  very  nearly  the  same  molecular  structure. 
This  is  the  case  with  many  substances  of  analogous, 
though  not  of  identical  composition,  which  are  called 
isomorphous.  Such  aggregates  are  therefore  called 
isomorphous  growths.  In  still  other  cases  the  crystals 
of  one  substance  may,  to  a  certain  extent,  affect  the 
orientation  of  other  crystals  with  a  different  molecular 
structure,  which  are  deposited  upon  them.  Such  ag- 
gregates are  called  regular  groivths,  to  distinguish  them 
from  those  in  which  the  arrangement  of  the  crystal 
individuals  is  entirely  irregular. 

The  six  possible  categories  of  crystal  aggregates 
may  therefore  be  classified  as  follows : 


182 


OR  TSTALLOGRAPHT. 


I.  HOMOGENEOUS  AGGREGATES.    Individuals  of  the  same 
substance. 

1.  Parallelism  complete.    .     .     Parallel  Growths. 

2.  Parallelism  partial Turin  Crystals. 

3.  Parallelism  wanting.      .  Irregular  Aggregates. 

II.  HETEROGENEOUS    AGGREGATES.     Individuals  of  dif- 
ferent substances. 

4.  Parallelism  nearly  complete. 

Isomorphous  Growths. 

5.  Parallelism  partial.    .     .     .     Regular  Groivtfjs. 

6.  Parallelism  wanting. 

Irregular  Heterogeneous  Aggregates. 

We  shall  now  consider  the  essential  characters  of 
these  six  groups  in  succession. 

I.  AGGREGATES  OF  CRYSTALS  OF  THE  SAME  SUBSTANCE. 

Parallel  Growths,     The  molecular  arrangement  must 
be  the  same  along  all  parallel  lines  within  a  crystal 


FIG.  285. 


FIG.  286. 


individual,  but  two  or  more  individuals  of  the  same 
substance  may  grow  side  by  side,  in  such  a  position 
that  their  molecular  arrangements  are  completely 
parallel  throughout.  Such  crystals  must  be  sym- 


CRYSTAL  AGGREGATES.  183 

metrical  with  reference  to  some  plane  which  is  a  plane  of 
symmetry  for  each  crystal  form.  They  may,  however, 
be  united  in  this  or  in  any  other  plane.  This  is 
shown  in  Figs.  285  and  286,  which  represent  two  octa- 
hedrons in  exactly  parallel  position,  and  therefore 
symmetrical  with  reference  to  the  faces  of  the  cube. 

In  the  first  case,  the  cubic  face  is  also  the  one  in 
which  the  two  individuals  are  united;  while  in  the 
second,  they  are  joined  in  an  octahedral  face,  though 
their  positions  are  still  exactly  parallel. 

A  large  number  of  individuals  may  be  united  in  this 
way  where  each  is  represented  by  only 
an  extremely  thin  lamella  (Fig.  287). 
This  results  in  the  alternate  repetition 
of  two  planes,  meeting  at  angles  which 
are  supplements  of  each  other. 

Strictly  speaking,  we  should  regard 
each  re-entering  angle  as  indicative  of 
a  separate  individual,  joined  to  the 
others  by  parallel  growth.  If,  how- 
ever, the  width  of  the  alternating  planes 
is  extremely  small,  the  separation  of 
the  different  individuals  becomes  very  slight,  and  the 
effect  is  of  a  single  crystal  whose  faces  are  finely 
striated.  In  such  cases  it  is  customary  to  speak  of 
the  alternating  planes  as  producing  an  oscillatory  com- 
bination of  forms  on  a  single  individual.  Striations 
caused  in  this  manner  are  among  the  most  frequent 
sources  of  imperfection  on  crystal  planes,  as  will  be 
more  fully  explained  in  the  succeeding  chapter. 

Parallel  growths  become  of  importance  in  connec- 
tion with  the  manner  in  which  many  substances  crys- 
tallize. When  the  rate  of  increase  is  rapid,  it  not 


184  CRYSTALLOGRAPHY. 

infrequently  happens  that  a  number  of  minute  crystals 
group  themselves  in  parallel  position  to  form  the 
skeleton  of  a  larger  crystal.  In  such  cases  the  smaller 
forms  have  been  called  by  Sadebeck  sub-individuals. 
A  familiar  example  of  this  mode  of  growth  is.  offered 
by  common  salt ;  and  the  same  thing  may  be  observed 
in  the  case  of  sulphur,  copper,  gold,  fluorspar,  quartz 
and  a  variety  of  other  substances. 

Twin  Crystals.  (German,  Zwillinge ;  French,  modes, 
hemitrop^  When  two  crystals  of  the  sanle  substance, 
or  two  halves  of  the  same  crystal,  are  not  in  completely 
parallel  position,  they  may  still  be  joined  in  such  a 
manner  that  some  crystallographic  plane,  or  at  least 
some  crystallographic  direction,  is  common  to  both. 
In  such  cases  the  two  crystals  or  halves  of  the  same 
crystal  are  symmetrical  with  reference  to  some  plane 
which  is  not  a  plane  of  symmetry  for  the  single  individuals. 
This  is  the  most  apparent  distinction  between  parallel 
growths  and  what  are  known  as  crystals  in  twinning 
position,  or  tivin  crystals. 

Symmetrical  aggregates  of  this  kind  are,  on  account 
of  their  complexity,  variety  and  frequency,  of  much 
importance  in  crystallography ;  and  are  therefore  de- 
serving of  particular  description. 

The  Twinning  Plane  and  Twinning  Axis.  The  relative 
position  of  two  crystals  in  twinning  position  may  be 
most  readily  understood  by  supposing  that  one  has 
been  revolved  through  180°  about  some  crystallo- 
graphic direction,  which  thus  remains  common  to 
both  individuals.  This  direction  or  line  of  revolution  is 
called  the  tivinning  axis,  and  it  is  in  most  cases  normal 
to  the  plane  with  reference  to  which  the  two  individ- 
uals become  symmetrical  after  the  revolution.  This 
plane  is  called  the  tivinning  plane. 


CRYSTAL  AGGREGATES.  185 

Suppose,  for  instance,  -^(jhat  we  imagine  an  octa- 
hedron cut  into  two  equal  parts  parallel  to  an  octa- 
hedral plane,  and  one  half  re- 
volved 180°  about  a  line  normal 
to  this  plane.  The  result  will  be 
the  position  shown  in  Fig.  288, 
where  the  plane  in  which  the  oc- 
tahedron is  divided  is  the  twin- 
ning plane,  and  the  line  normal 
to  it  is  the  twinning  axis. 

If,   however,   the    octahedron 
were  composed  of  a  positive  and  (Spinel.) 

negative  tetrahedron  in  equal  development,  the  two 
halves,  after  such  a  revolution,  would  not  be  symmet- 
rical with  reference  to  this  plane,  because  a  negative 
face  would  lie  over  a  positive  face,  and  vice  versa. 
The  two  halves  would  then  be  symmetrical  with  refer- 
ence to  a  face  of  the  icositetrahedron,  202,  {211}. 
Such  instances,  where  the  twinning  plane  is  not 
normal  to  the  twinning  axis,  are  not  common. 

Any  crystallographic  plane  not  a  plane  of  symmetry 
may  be  a  twinning  plane,  but  the  most  frequent  twin- 
ning planes  in  all  systems  are  those  which  possess  the 
simplest  indices. 

Contact  Twins  and  Composition  Face.  Two  individual 
crystals  in  twinning  position  are  usually,  though  by  no 
means  always,  united  in  a  plane,  which  may  or  may 
not  coincide  with  the  twinning  plane.  Twins  of  this 
sort  are  called  contact  or  juxtaposition  twins,  and  the 
plane  in  which  they  are  in  contact  is  called  their  com- 
position face.  This,  like  the  twinning  plane,  is  gener- 
ally a  plane  that  has  very  simple  indices. 

Figs.  289  and  290  represent  twins  of  two  monoclinic 


186 


CR  TSTALLOORAPHT. 


(Gypsum.) 


FIG.  290. 
(Orthoclase.) 


minerals,  gypsum  and  orthoclase,  in  both  of  which 
the  twinning  plane  is  the  orthopinacoid. 
In  the  first  this  is  also  the  composition 
face,  while  in  the  second  the  composi- 
tion face  is  at  right  angles  tovits  twin- 
ning plane,  and  is  itself 
the  plane  of  symmetry, 
which  could  not  there- 
fore be  a  twinning  plane. 
Both  twinning  plane 
and  composition  face 
are  crystallographically 
possible  planes,  except  in  the  mono- 
clinic  and  triclinic  systems.  Here  they 
may  be  surfaces  which  are  impossible 
as  crystal  planes,  but  they  are  either 
(1)  perpendicular  to  a  possible  crystal  edge,  or  (2)  per- 
pendicular to  a  possible  face  and  parallel  to  a  possible 
edge. 

Penetration  Twins.  It  happens  quite  frequently  that 
two  crystals  in  twinning  position  are  not  joined  in  a 
single  plane,  but  that  there  is  a  com- 
plete interpenetration  of  both  indi- 
viduals. This  is  shown  in  Fig.  291, 
which  represents  two  twin  rhombo- 
hedrons,  symmetrical  with  reference 
to  a  prism  of  the  first  order,  but 
without  any  composition  face.  There 
FIG.  291.  ig  a  complete  interpenetration  of 
(Hematite.)  their  substance,  and  the  space  com- 
mon to  both  is  very  irregularly  distributed  between 
them,  as  may  be  shown  by  an  examination  in  polarized 
light.  Only  those  portions  of  each  crystal  which  pro- 
ject beyond  the  limits  of  the  other  possess  a  uniform 


CRYSTAL  AGGREGATES. 


187 


molecular  structure.     Twin  crystals  of  this  kind  are 
called  penetration  tivins. 

Supplementary  Twins.  Partial  forms  (hemihedral, 
tetartohedral  or  hemimorphic)  have  lost  more  or  less 
of  the  symmetry  belonging  to  their  corresponding 
holohedrons,  and  may  consequently  possess  twinning 
planes  which  are  impossible  for  the  latter.  These 
partial  forms  frequently  form  twins  with  the  axes  par- 
allel for  both  individuals,  which  would,  of  course,  be 
impossible  for  holohedral  crystals.  They  are  called 
supplementary  twins  (German,  Erganzungszwillinge), 
since  the  union,  especially  by  penetration,  of  two 
hemihedrons  in  this  manner,  tends  to  restore  the  lost 


FIG.  292. 
(Tetrahedrite.) 


FIG.  293. 
(Calamine.) 


holohedral  symmetry.  Fig.  292  shows  two  tetrahedrons 
with  parallel  axes,  and  symmetrically  placed  with 
reference  to  the  faces  of  the  cube  (tetrahedrite  and 
diamond).  Fig.  293  is  another  case  of  a  supplementary 
twin,  formed  from  two  hemimorphic  crystals,  placed 
symmetrically  to  their  basal  pinacoid  (calamine). 

Repeated  Twinning.  To  the  second  individual  of  a 
twin,  a  third  may  be  placed  in  twinning  position  ac- 
cording to  the  same  law  as  the  first.  This  produces 
a  trilling  (German,  Drilling).  Four  individuals  related 
in  this  way  make  a  fourling  (German,  Vierling\  etc. 
Such  a  repetition  of  twinning  according  to  the  same 


188 


CR  T8TALLOGRAPHT. 


law  may  take  place  by  either  of  two  methods  :  (1)  the 
twinning  plane  may  remain  parallel  to  itself,  so  that  the 

alternate  individuals  are 
in  parallel  position ;  or 
(2)  the  twinning  plane 
may  change  its  direction, 
as  when  the  two  symmet- 
rical faces  of  a  prism  suc- 


\ 


/fx?Yl/2X 

ii 

HI 

///     /// 

p  P 

6 

k 

b. 

f      f 

\  J 

7 

I 

s 

X  / 

cessively  serve  as  twin- 


FIG.  294. 
(Aragonite.) 


ning   planes.     Figs.  294 
and  295  illustrate  these 


m 


FIG.  296. 
(Rutile.) 


Fio.  295. 
(Aragonite.) 

two  methods  of  repeated  twinning,  where,  in  both  cases, 
the  unit  prism  is  the  twinning  plane.  The  results  of 
the  first  method  are  called  poly  synthetic  twins;  and  of 
the  second,  cyclic  tivins  (Ger- 
man, Wendezivillinge),  on  ac- 
count of  their  tendency  to  pro- 
duce circular  groups,  as  in  the 
case  of  rutile  (Fig.  296).  Such 
cyclic  groups  are  more  or  less 
symmetrical,  according  as  the 
angle  between  the  successive  sets  of  axes  is  more  or 
less  exactly  a  divisor  of  360°. 

Compound  Twins.  These  are  produced  by  the  presence 
of  two  or  more  twinning  laws  in  the  same 
group.  An  instance  of  this  is  shown  in 
Fig.  297,  where  two  orthoclase  twins,  like 
that  represented  in  Fig.  290,  are  again 
.twinned  parallel  to  their  basal  pinacoid. 
Compound  twinning  may  lead  to  very  com- 
I  plicated  relations.  Comparatively  simple 
examples  of  it  are  found  on  crystals  of  the 
minerals  marcasite,  chalcocite,  albite,  stau- 
rolite,  and  tridymite. 


FIG.  297. 
(Orthoclase.) 


CRYSTAL  AGGREGATES.  189 

Mimicry.  The  general  tendency  of  twinning  is  to 
increase  the  grade  of  symmetry.  Three  orthorhom- 
bic  individuals,  the  angle  between 
whose  symmetrical  twinning  planes 
is  nearly  120°,  may  interpenetrate 
so  as  to  form  an  almost  perfectly 
hexagonal  combination.  This  is 
the  case  with  witherite  (BaCO3), 
the  angle  between  whose  twinning 

planes,    OOPAOOP,    {110}    A   {110},  (Chrysoberyl.) 

is  118°  30' ;  and  with  chrysoberyl,  the  angle  between 
whose  twinning  planes,  Poo  A  P&,  {011}  A  {Oil},  is 
119°  46'  (Fig.  298).  This  group  can  also  be  explained 
by  assuming  the  brachydome,  3Pob,  {031(  as  twinning 
plane,  whose  interfacial  angle  3Pob  A  3Pdb,  {031}  A 
{031},  is  59°  46'. 

Pseudosymmetrical  crystals  (p.  92),  that  is,  such  as 
closely  simulate  a  higher  symmetry  than  they  really 
possess,  are  enabled  by  repeated  twinning  to  greatly 
increase  their  deceptive  appearance.  In  many  in- 
stances, their  true  character  can  only  be  determined 
by  optical  means.  This  phenomenon  has  been  called 
by  Tschermak  mimicry  (German,  Mimesie).  Examples 
of  it  are  offered  by  the  pseudo-monoclinic  microcline 
and  the  pseudo-rhombohedral  chabazite,  which  are 
really  triclinic ;  by  the  pseudo-tetragonal  apophyllite, 
which  is  monoclinic ;  and  by  the  pseudo-isometric 
leucite,  which,  at  ordinary  temperatures,  is  orthorhom- 
bic.  Harmotome  and  phillipsite  are  monoclinic,  but 
they  possess  interfacial  angles  which  allow  them,  by 
repeated  twinning,  to  closely  approach  a  tetragonal,  or 
even  an  isometric  habit.  This  property  of  mimicry  is 
of  importance  in  explaining  many  of  the  so-called 
"  optical  anomalies  "  exhibited  by  crystals. 


190  CRYSTALLOGRAPHY. 

Mode  of  Formation  of  Twin  Crystals,  As  has  been 
already  stated  in  Chapter  I,  the  perfection  of  crystal- 
lization in  a  given  substance  is  inversely  proportional 
to  the  rapidity  of  its  solidification.  If  the  molecular 
forces  have  full  time  to  act,  the  parallelism  of  the 
molecules  will  be  complete.  In  case  the  hypothesis 
that  twin  crystals  are  due  to  a  partial  parallelism  of  the 
molecules  is  correct,  we  might  expect  to  produce  them 
by  artificially  retarding  the  motion  of  the  molecules 
at  the  time  of  solidification.  This  has  actually  been 
accomplished  by  O.  Lehmann,  who  found  that  barium 
chloride  and  some  other  salts,  which  habitually  pro- 
duce simple  forms  on  crystallizing  from  an  aqueous 
solution,  appeared  in  twins  when  such  a  solution  was 
mixed  with  gum  or  some  other  viscous  substance  which 
retarded  molecular  movement.* 

In  the  case  of  contact  twins,  the  hemitropic  charac- 
ter appears  to  date  from  the  first  inception  of  the 
crystal,  as  is  shown  by  the  fact  that  the  most  minute 
individuals  observable  with  the  microscope  are  as  per- 
fect twins  as  those  of  large  size.  When  such  a  double 
crystal  has  once  been  started,  its  growth  is  regular  in 
both  directions  away  from  the  composition  face.  The 
result  of  this  is  a  general  shortening  of  each  individual 
in  the  line  of  the  twinning  axis,  and  the  frequent  pro- 
duction of  two  half-crystals  in  twinning  position. 

Penetration  or  repeated  twins,  on  the  other  hand, 
show  a  constant  tendency  to  the  addition  of  layers  in 
hemitropic  position  with  reference  to  those  which  pre- 
cede. This  may  be  due  to  the  impurity  of  solution  or 
viscosity  of  the  magma  in  which  crystallization  is  tak- 
ing place. 

*  Molecularphysik,  vol.  i.  p.  415. 


CRYSTAL  AGGREGATES.  191 

Mimetic  twins  may  be  produced  in  dimorphous  sub- 
stances by  physical  conditions  unsuited  for  the  exist- 
ence of  the  molecular  arrangement  with  which  they 
solidified.  The  new  conditions  may  produce  a  new 
molecular  structure,  which,  by  twinning,  is  made  to 
fit  into  the  original  crystal  form.  This  is  the  case 
with  boracite,  tridymite  and  leucite.  When  the  origi- 
nal conditions  are  restored  (generally  by  raising  the 
temperature)  the  internal  structure  again  comes  into 
accord  with  the  external  form. 

Examples  of  Common  Twinning  Laws.*  It  will  be 
found  advantageous  to  briefly  explain  a  number  of 
figures  which  represent  concrete  examples  of  the  com- 
monest modes  of  twinning  in  each  of  the  six  crystal 
systems. 

In  the  isometric  system,  a  large  majority  of  all  twins 
are  formed  according  to  the  law  illustrated  by  Fig. 
288  (p.  185),  where  the  twinning  plane  is  an  octahe- 
dral face,  and  the  twinning  axis  a  normal  to  this. 
This  law,  commonly  called  the  "  spinel  law,"  from  its 
frequent  occurrence  on  crystals  of  the  aluminate, 
spinel,  is  capable  of  producing  a  great  variety  of  re- 
sults which  differ  according  to  the  habit  of  the  crys- 
tals which  are  twinned,  as  well  as  according  to  whether 
contact  or  penetration  twins  are  formed.  Figs.  299, 
300  and  301  show  contact  twins  of  the  three  simplest 
isometric  holohedrons  according  to  this  law ;  while 
Figs.  302,  303  and  304  represent  penetration  twins  of 
the  same  forms  by  the  same  method.  Fig.  305  shows 

*  For  a  more  complete  illustration  of  this  subject  see  Rose-Sade- 
beck's  Crystallography,  vol.  u  (1876) ;  Klein's  Dissertation  on  Crys- 
tal Twinning  and  Distortion  (1876) ;  and  E.  S.  Dana's  Text-book  of 
Mineralogy. 


192  CRYSTALLOGRAPHY. 

a  penetration  twin  of  two  tetrahedrons,  also  symmet- 


FIG.  299. 
(Copper.) 


FIG.  300. 
(Spinel.) 


FIG.  301. 
(Ziuc  blende.) 


rical  to  the  octanedral  face ;  while  another  penetra- 
tion twin  of  the  same  two  forms  symmetrical  to  the 


FIG.  302. 
(Fluorspar.) 


Fio.  303. 
(Spinel.) 


face  of  the  cube  (which  is  no  longer  a  plane  of  sym- 
metry in  these  figures)  is  shown  in  Fig.  292  on  p.  187. 


FIG.  305. 
(Tetrahedrite.) 


FIG.  306. 
(Pyrite.) 


A   penetration  twin  of  two   pentagonal   dodecahe- 
drons (known  as  the  iron  cross)  whose  twinning  plane 


CRYSTAL  AGGREGATES. 


193 


is  the  rhombic  dodecahedron,  GO  0,  1110},  is  shown  in 
Fig.  306. 

In  the  tetragonal  system  the  unit  pyramid  of  the 
second  order,  Poo,  jOllj,  is  the  most  common  twin- 
ning plane.  This  is  represented  in  Fig.  307,  as  it 
occurs  on  crystals  of  zircon  (ZrSiO4),  bounded  by  the 
forms  oo  P,  \UO\  (m);  3P,  {331}  (u) ;  2P,  J221J  (p); 
and  P,  Jill |  (o).  The  same  law 
is  common  on  cassiterite  (SnO2), 
and  on  rutile  (TiO2).  On  crys- 
tals of  the  latter  mineral  repeated 
twinning  is  common.  This  is 
accomplished  in  two  ways  where 
cyclic  twinning  results.  In  one 
case,  opposite  faces  of  the  pyra- 
mid, (Oil)  and  (Oil),  alternately  serve  as  twinning 
planes.  This  keeps  the  vertical  axes  in  the  same  plane, 
and,  as  they  diverge  at  angles  of  65°  35',  six  individuals 
approximately  complete  the  circuit,  as  shown  in  Fig. 
296  (p.  188).  In  other  cases,  contiguous  faces  of  the 
same  pyramid  (Oil)  and  (101)  alternately  serve  as  twin- 
ning planes.  This  causes  the  vertical  axes  to  form  a 
zigzag,  and  eight  individuals  are  necessary  to  com- 


FIG.  307. 
(Zircon.) 


FIG.  309. 
(Rutile.) 

Rutile  twins  of  this  kind  from 


plete  360°  (Fig.  308). 

Graves  Mi,  Ga.,  often  have  their  prism  faces  of  the 


194 


CRYSTALLOGRAPHY. 


second  order,  GO  P  GO,  J  010 }  (a),  extended  to  intersec- 
tion, which  produces  a  tetragonal  scalenohedron,  as 
shown  in  Fig.  309. 

In  the   tetragonal  mineral   chalcopyrite  v(FeCuS9), 


FIG.  310.  FIG.  311. 

(Calcite.)  (Calcite.) 

which  approaches  very  closely  to  an  isometric  crystal- 
lization (p.  92),  the  twinning  plane  is  the  unit  pyramid, 
P,  jlll|.  This  corresponds  to  the  spinel  law  in  the 
isometric  system  (p.  191).  The  tetragonal  but  pyram- 


FIG.  312. 
(Calcite.) 


FIG.  313. 
(Calcite.) 


idally  hemihedral  scheelite  (CaWO4)  forms  supple- 
mentary twins,  where  the  twinning  plane  is  the  unit 
prism,  oo  P,  {HO^. 

In  the  hexagonal  system  holohedral  substances 
and  consequently  holohedral  twins  are  rare.  On 
pyrrhotite,  magnetic  pyrite  (Fe7S8),  the  unit  pyramid, 


CRYSTAL  AGGREGATES. 


195 


P,  {lllf,  has  been  observed  as  twinning  plane;  and 
on  tridymite  (SiOa),  the  two  pyramids,  ^P,  J1016J  and 
JP,  J3034[.  On  rhombohedral  crystals  twinning  is 
very  common,  particularly  so  on  the  best  represent- 
ative of  this  group,  calcite  (CaCO3).  Here  we  have  as 
twinning  plane  sometimes  the  basal  pinacoid  with  the 
vertical  axis  as  twinning  axis  (Figs.  310  and  311) ; 
sometimes  the  negative  rhombohe- 
dral face,  —  fff,  A:  j  0112}  (Fig.  312); 
sometimes  the  positive  rhombohe- 
dron,  Rj  /cjlOllJ,  bringing  the  two 
vertical  axes  nearly  at  right  angles 
to  one  another  (Fig.  313) ;  and 
sometimes  the  rhombohedron,  —  2P, 
/<•  j  0221}  (Fig.  314).  A  penetration 
twin  of  two  rhombohedrons,  as  some- 
times seen  on  crystals  of  ferric  oxide, 
hematite,  is  shown  in  Fig.  291  (p.  186). 

Tetartohedral  hexagonal  forms,  like  those  occurring 
on  crystals  of  quartz,  produce  supplementary  twins, 
which  tend  vto  restore  a  higher  grade  of  symmetry. 
Thus  a  complete  interpenetration  of  two  right-handed 
or  of  two  left-handed  individuals,  one  of  which  has  been 

revolved  180°  (or  60°)  about 
its  vertical  axis,  reproduces 
a  trapezohedral  hemihe- 
dral  form  (Dauphine  law, 
Fig.  315) ;  while  a  similar 
interpenetration  of  a  right- 
and  a  left-handed  crystal 
restores  a  scalenohedral 
symmetry,  with  the  prism 
of  the  second  order,  oo  P2, 
/CTJ1120S  as  twinning  plane  (Brazilian  law,  Fig.  316). 


FIG.  315. 
(Quartz.) 


Fm.  316. 
(Quartz.) 


196 


CR  YSTALLOORAPHY. 


Contact-twins  of  quartz,  with  the  individuals  symmet- 
rical to  both  oo  P,  {1010}  and  R,  Arr{10ll}  also  occur. 
In  the  orthorhombic  system  the  most  common  twin- 
ning plane  is  the  unit  prism  oo  P,  {110},  as  exemplified 
by  both  the  polysynthetic  and  cyclic 
twins  of  aragonite  shown  in  Figs.  294 
and  295  (p.  188).     Tabular  crystals  of 
lead    carbonate    (cerussite)    bounded 
by  the  forms  ooPob,  {010}  (6);    oo  P, 
|110 1  (m) ;  and  P,  {111}  (p)  (Fig.  317), 
form  groups  in  which  both  x  P,  j  110  \ 
and  oo  P3,  {  130  \  act  as  twinning  planes. 
Orthorhombic  iron  disulphide  (mar- 
casite)  sometimes  shows  cyclic  groups 

FIG.  317.  i;    />          •      -i  •    •  -i       i        i  i-i-i 

(Cerussite.)  of  five  individuals,  bounded  by  oo  P, 
{110}  (M);  P3b,  {011}  (I),  and  OP,  {001},  and  united 
into  a  pentagonal  figure  by  oo  P,  {110}  (If)  as  twin- 
ning plane  (Fig.  318).  The  closely  related  iron  sulph- 
arsenide  (arsenopyrite),  bounded  byooP,  {110}  (M), 


FIG.  318. 
(Marcasite.) 


FIG.  319. 
(Arsenopyrite.) 


and  ^Pob,  { 013 }  (r),  forms  penetration  twins  of  two  in- 
dividuals, where  Poo,  { 101 }  acts  as  twinning  plane 
(Fig.  319). 

The  group   of   chrysoberyl  crystals  shown  in  Fig. 
298  (p.  189)  are  united  with  3Pob,  {031}  as  twinning 


CRYSTAL  AGGREGATES. 


197 


plane.  The  copper  sulphide,  chalcocite,  forms  three 
kinds  of  twins  symmetrical  to  oo  P, 
1 110},  to  fP&,  J043},  and  to  £P, 
J112J  respectively.  The  latter  is 
shown  in  Fig.  320.  The  ortho- 
rhombic  silicate,  staurolite,  which 
is  commonly  bounded  by  the 
planes  oo  P&,  {010}  (o);  OP,  |001} 
(P);ooP,  |110}  (Jf);  and  Pco  {101} 
(r),  forms  rectangular  crosses  with  (Chalcocite.) 

fPoo,    {032}  as  twinning   plane   (Fig.  321),  and   also 


Fio.  322. 
(Staurolite.) 

oblique  crosses  with  f  P  f  { 232 }  acting  in  the  same 
capacity  (Fig.  322). 

In  the   monoclinic   system  any  face  may  serve  as 
twinning  plane  except  the  clinopinacoid, 
which  is  the  plane  of  symmetry.     Even 
this,  however,  is  not  infrequently  a  com- 
position face  for  contact  twins,  as  in  the 
case  of  orthoclase  (Fig.    290,  p.    186). 
The  most  common  twinning  plane  is  the 
orthopinacoid  oo  Pco,  { 100 } ,  as  may  be 
seen  in  the  case  of  gypsum  (Fig.  289,  p.          Fio  g23 
186)  and  malachite  (Fig.  323).    The  same        (Malachite.) 
law  is  also  exemplified  in  augite,  hornblende,  feld- 


m 


m 


- 


198 


CR  YSTALLOGRAPHY. 


spar,  and  epidote.  A  twin  of  the  latter  mineral 
with  the  forms  OP,  { 001}  (M)\ 
oo  Poo,  {1001  (T);  Pob,jlOl}(r); 
oo  Poo  |010}  (P);+P,  SlU}  (n); 
oo  P2,  1210}  (M);  vand  £  Poo, 
|012J  (&),  is  shown  in  Fig.  324. 
The  monoclinic  feldspar,  ortho- 
clase,  exhibits  a  number  of  dif- 
ferent  twinning  laws.  In  addi- 
tion to  the  one  explained  on  p.  186  (Carlsbad  law), 
the  twinning  plane  is  sometimes  the  basal  piriacoid 


FIG.  325. 
(Orthoclase.) 

(Mannebacher  law) ;  and  sometimes  a  clinodome  whose 
symbol  is  2Poo  ,  { 021 }  (Baveno  law).  Contact  twins 
formed  according  to  these  two  laws 
are  shown  in  Figs.  325  and  326.  The 
silicate,  augite,  sometimes  has  as 
twinning  plane  the  clinopyramid 
P2,  {122}  (Fig.  327).  The  monoclinic 
mica  differs  but  slightly  from  a  hex- 
agonal mineral  in  its  symmetry  and 
angles.  Its  twinning  axis  is  often  a 
line  in  the  basal  pinacoid,  normal  to 
the  combination  edge  OP,  {001}  :  oo  P,  jllOj.  The 


FIG.  327. 
(Augite.) 


CRYSTAL  AGGREGATES. 


199 


twinning  plane  is  therefore  a  crystallographically  im- 
possible  face,   normal   to   the 
basal  pinacoid ;  while  the  com- 
position face  is  generally  the 
basal  plane  (Fig.  328). 

In  the  triclinic   system  any 
face  may  act  as  twinning  plane, 
or  may  be,  and  frequently  is 
one  which,  on  account  of  its  irrational  indices,  is  not 
possible  as  a  crystal  plane.     On  crystals  of  the  soda 


FIG.  328. 
(Mica.) 


FIG.  329.  FIG.  330. 

(Albite.)  (Albite.) 

feldspar,  albite,  we  find  the  brachypinacoid  as  twin- 
ning   plane   (Fig.    329),  and    also    the   brachydome, 


FIG.  331.  FIG.  332. 

(Albite.)  (Albite.) 

^Pob'  |021}  (Fig.  33C).  The  analogue  of  the  Manne- 
bacher  law  occurs  on  the  triclinic  feldspars  when  the 
twinning  axis  is  the  macrodiagonal,  and  the  twinning 


200  CRYSTALLOGRAPHY. 

plane  normal  to  this,  and  therefore  no  possible  crystal 
plane  (Pericline  law,  Fig.  331).  To  avoid  the  re-en- 
trant angles  produced  by  this  law  when  the  basal 
pinacoid  is  the  composition  face,  the  two  individuals 
are  generally  united  in  that  particular  rhombic  section 
(German,  rhombischer  Schnitt)  which  is  common  to 
both  (Fig.  332).  The  exact  position  of  this  section,  of 
course,  depends  on  the  relative  inclinations  of  the 
axes.  A  variety  of  twinning  laws,  where  the  twinning 
planes  are  not  crystallographically  possible  faces,  have 
also  been  observed  on  the  triclinic  orthosilicate  of 
alumina,  Al2SiO5  (cyanite,  disthene). 

Irregular  Homogeneous  Aggregates  of  Crystals.  When 
crystallization  begins  simultaneously  at  many  separa- 
ted points  in  a  saturated  solution  or  cooling  magma, 
individual  crystals  are  formed  whose  molecular  struc- 
tures are  entirely  independent  in  their  orientation, 
although  they  may  be  identical  in  their  nature.  If 
such  crystals  continue  to  grow  until  they  come  in  con- 
tact, a  wholly  irregular  aggregate  is  the  result  (Figs. 
21  and  22,  p.  17).  Such  aggregates  are  classified  ac- 
cording to  the  extent  to  which  their  crystals  are  indi- 
vidually developed  (crystal  and  crystalline  aggre- 
gates) ;  according  to  their  texture  (porous  or  compact) ; 
and  according  to  the  size  of  their  grain  (coarse  or  fine). 

Other  crystalline  aggregates  are  not  entirely  irreg- 
ular in  the  arrangement  of  their  component  individ- 
uals. Certain  acicular  crystals  have  a  tendency  to 
arrange  themselves  radially  about  a  point,  so  as  to 
form  spherulitic  groups  (wavellite,  stilbite),  while 
others  have  a  similar  radial  arrangement  about  a  line 
(aragonite,  "  flos  ferri ").  Many  fibrous  crystals  group 
themselves  with  their  long  axes  nearly  parallel  (gyp- 


CRYSTAL  AGGREGATES.  201 

sum,  chrysotile,  asbestos).  Tabular  or  scaly  crystals 
form  lamellar  or  foliated  aggregates  (wollastonite, 
brucite,  gypsum,  mica). 

These  types  of  aggregates  produce  an  almost  end- 
less number  of  varieties  and  compound  forms,  whose 
special  description  must  be  sought  in  a  larger  work. 

In  still  other  cases  there  is  a  near  approach  to  par- 
allelism in  the  individual  crystal  forming  the  group. 
Examples  of  this  may  be  found  in  the  so-called  "  iron- 
rose  "  (hematite)  of  Switzerland ;  in  stilbite,  and  in  the 
twisted  quartz  crystals  from  Switzerland. 

II.  AGGREGATES  OF  CRYSTALS  OF  DIFFERENT  SUBSTANCES. 

Isomorphous  Growths.  It  was  long  ago  observed  by 
Mitscherlich  that  substances  of  analogous  chemical 
composition  were  apt  to  possess  very  similar  crystal 
forms.  Such  substances  he  called  isomorphous.  The 
shape,  size  and  mode  of  arrangement  of  their  j^ivsical 
molecules  must  be  nearly  alike,  since  the  ^1  •  iso- 
morphism is  the  ability  of  molecules  of  tw^P^nore 
substances  to  enter  indiscriminately  into  the  formation 
of  a  single  crystal ;  or  at  least  the  ability  of  a  crystal 
of  one  substance  to  continue  its  growth  in  a  saturated 
solution  of  another.  If  we  suspend  a  crystal  of  chro- 
mium alum  in  a  solution  of  potash  alum,  it  will  soon 
be  coated  with  a  transparent  layer  of  the  latter  salt, 
which  perfectly  preserves  the  form  of  the  original 
crystal.  This  may  be  again  covered  with  a  similar 
layer  of  chromium  alum,  and  so  on  to  any  number  of 
successive  zones  whose  physical  molecules  are  all  in 
parallel  orientation.  The  same  thing  takes  place  when 
a  crystal  of  calcium  carbonate  is  hung  in  a  solution 
of  sodium  nitrate.  Such  concentric  zones  of  differ- 


202       •  CRYSTALLOGRAPHY. 

ent  substances   with   completely   parallel   molecular 
structures  are  of  frequent  occurrence  in  nature,  but 

they  are  only  to  be  found 
amongvisomorphous  com- 
pounds, like  the  garnets, 
tourmalines,  micas,  pyrox- 
enes, feldspars,  etc.  A 
striking  example  is  the 
parallel  growth  between  crystals  of  the  isomorphous 
xenotime  (YPO4)  and  malacon  (ZrSiO4+  aq).  Another 
is  the  growth  of  a  zone  of  epidote  around  the  iso- 
morphous silicate,  allanite  (Fig.  333). 

Regular  Growths  of  Different  Minerals.  It  is  worthy 
of  note  that  a  certain  regularity  of  arrangement  exists 
between  the  crystals  of  substances  which  are  alto- 
gether unlike  in  chemical  composition.  For  instance, 
the  triclinic  cyanite  and  the  orthorhombic  staurolite 
grow  together  so  that  their  crystals  have  one  face  and 
one  axis  in  common.  The  tetragonal  rutile  (TiOa) 
grows  upon  the  r-hombohedral  titanic  iron  (FeTiO3),  so 
that  its  prism  of  the  second  order,  oo  Poo  (100),  coin- 
cides with  the  basal  pinacoid  of 
the  latter  mineral,  while  its  ver- 
tical axis  has  the  direction  of  one 
of  the  intermediate  lateral  axes 
of  the  iron  ore  (Fig.  334).  Sim- 
ilar examples  of  partial  orienta- 
tion are  to  be  found  between 
quartz  and  calcite  ;  between  chal- 
copyrite  and  tetrahedrite ;  and  between  magnoferrite 
and  hematite. 

They  also  exist  between  dimorphous  modifications 
of  the  same  substance,  as  marcasite  and  pyrite,  calcite 


CRYSTAL  AGGREGATES.  203 

and  aragonite,  pyroxene  and  hornblende,  etc.,  where 
they  have  originated,  in  some  cases  at  least,  by  the 
partial  alteration  of  one  modification  into  the  other. 

In  other  cases,  the  crystals  so  related  have  analo- 
gous compositions  and  crystallize  in  different  systems, 
but  with  very  similar  forms,  as  orthoclase  and  albite, 
orthorhombic  and  monoclinic  pyroxene,  etc. 

A  partial  parallelism  in  orientation  is  also  frequent 
between  the  inclusions  in  many  crystals  and  their 
host,  as  for  instance,  rutile  in  mica,  augite  in  leucite, 
hematite  in  feldspar,  coaly  matter  in  andalusite  (chi- 
astolite),  etc.  This  subject  becomes  of  considerable 
importance  in  microscopical  mineralogy  and  petrog^ 
raphy.* 

Irregular  Heterogeneous  Growths  of  Crystals.  This  is 
the  most  general  form  of  grouping  possible.  It  is  purely 
accidental  and  obeys  no  rules,  although  a  thorough 
understanding  of  it  implies  a  knowledge  of  the  laws  of 
association  and  paragenesis,  which  are  of  great  im- 
portance in  the  mineral  world.  The  study  of  hetero- 
geneous crystal  aggregates  belongs,  however,  rather 
to  the  domain  of  petrography  than  to  that  of  miner- 
alogy or  crystallography. 

*  For  full  description  and  list  of  regular  growths  of  minerals 
of  different  species,  both  isomorphous  and  otherwise,  see  Sadebeck: 
Angewandte  Krystallographie,  pp.  244-249;  anclO.  Lehmann:  Mole- 
cularphysik,  vol.  i.  pp.  293-407.  The  latter  work  cites  numerous 
instances  f  ro.n  artificial  salts. 


CHAPTEE  X. 

IMPERFECTIONS  OF  CRYSTALS. 

Sources  of  Imperfection.  We  have  thus  far  considered 
crystals  in  their  ideal  development ;  that  is,  as  sym- 
metrical polyhedrons,  bounded  by  mathematically 
plane  surfaces  intersecting  at  fixed  angles.  If  the 
molecular  forces  of  a  single  substance  at  the  time  of 
its  solidification  were  entirely  free  to  act,  without  hin- 
drance of  any  kind ;  and  if  the  crystal  growth  could 
proceed  with  great  slowness  and  perfect  regularity  in 
all  directions,  such  ideal  crystal  forms  would  un- 
doubtedly result.  So  sensitive,  however,  are  these 
forces  to  obstacles  of  many  kinds  that  it  is  only  in  rare 
cases,  either  in  nature  or  the  laboratory,  that  they 
succeed  in  accomplishing  the  most  perfect  result  of 
which  they  are  capable. 

An  acquaintance  with  the  theoretical  crystal  form 
and  surface  may  be  obtained  from  figures  and  models 
constructed  so  as  to  show  them  in  their  ideal  perfec- 
tion. Natural  crystals,  however,  rarely  attain  such 
perfection  of  development,  and  a  knowledge  of  some 
of  the  causes  which  render  them  more  or  less  incom- 
plete is  necessary  for  their  thorough  understanding. 

Crystals  may  fall  short  of  perfection  by — 

1.  Distortion  of  form. 

2.  Irregularity  of  planes  or  angles. 

3.  Internal  impurity. 

304 


IMPERFECTIONS  OF  CRYSTALS.  205 

Distortion  of  Crystal  Form.  The  most  common  source 
of  this  is  unequal  rapidity  of  crystal  growth  in  different 
directions.  This  may  result  in  a  disguising  of  the 
real  form,  as  has  been  already  described  and  illustrated 
in  Chapter  I.  When  carried  to  excess,  certain  planes 
of  a  form  are  altogether  crowded  out  (merohedrism, 
p.  39),  which  often  produces  close  imitations  of  shapes 
characteristic  of  other  systems.*  Instances  of  such 
distortion  are  particularly  frequent  in  the  isometric 
system,  where  elongation  in  the  direction  of  a  principal 
axis  produces  a  tetragonal;  in  the  direction  of  a 
trigonal  axis  (p.  47),  a  rhombohedral ;  and  in  the 
direction  of  a  digonal  axis,  an  orthorhombic  appear- 
ance. This  may  be  seen  from  the  three  following 
figures  of  the  rhombic  dodecahedron  distorted  in  these 
three  directions  (Figs.  335,  336  and  337).  Fig.  338 
gives  the  result  of  an  elorgation  of  the  icositetrahedron, 
2  02,  1 211 }  in  the  direction  of  the  trigonal  axis. 


FIG.  335.  FIG.  336.  FIG.  337.  FIG.  338. 

When  accompanied  by  merohedrism,  which  generally 
obeys  some  definite  law,  surprising  results  are  some- 
times produced  by  the  distortion  of  isometric  forms. 

*•  This  phenomenon  has  been  called  pseudosymmetry  by  Sadebeck, 
who  used  the  terra  in  a  sense  very  different  from  that  in  which  it 
is  employed  by  Tschermak  (see  p.  92).  In  the  former  usage  the 
imitation  is  always  of  a  lower,  while  in  the  latter  it  is  of  a  higher 
grade  of  symmetry. 


206  CRYSTALLOGRAPHY. 

The  octahedron,  by  the  crowding  out  of  two  of  its  op- 
posite faces,  may  become  a  rhombohedron  (Fig.  339). 
Certain  crystals  of  green  fluorspar  from  Saxony,  show- 
ing the  form  GO  03,  {310J,  have  one  half  of  their  planes 
developed  at  the  expense  of  the  other  half,  so  as  to 
produce  a  hexagonal  scalenohedron  (Fig.  340).  The 
icositetrahedron  is  peculiarly  liable  to  such  distortion. 


FIG.  339.  FIG.  340.  FIG.  341.  FIG.  342. 

(Magnetite.)  (Fluorspar.)  (Gold.)      (Sal  ammoniac.) 

Gold  crystals  of  the  form  303,    {311},  sometimes  re- 
semble combinations   of  rhombohedron  and  scaleno- 


FIG.  343.  FIG.  344. 

(Pyrite.)  (Potassium  chloride.) 

hedron  (Fig.  341).  The  same  form  on  ammonium 
chloride  may  have  only  six  of  its  twenty- four  planes 
developed,  and  in  this  way  give  rise  to  a  form  resem- 
bling a  tetragonal  trapezohedron  (Fig.  342). 

It  has  been  observed  that  the  pentagonal  dodecahe- 

r<x>  021 

dron,  — £ —  ,  7t  \  201 } ,  on  pyrite  may  have  but  six  of 
its  twelve  planes  developed  in  such  a  manner  as  to  pro- 


IMPERFECTIONS  OF  CRYSTALS.  207 

duce  an  apparent  rhombohedron  (Fig.  343)  ;*  and  the 
icositetraliedron,  404,  {411},  on  potassium  chloride 
sometimes  produces  a  similar  result  by  the  survival 
of  only  one  fourth  of  its  planes.  In  fact,  two  such 
rhombohedrons,  in  apparent  twinning  position,  may 
be  derived  from  the  planes  of  the  same  icositetraliedron 
(Fig.  344).f 

Fig.  345  represents  a  crystal  of  iron  pyrites  showing 
representatives  of  all  the  forms  of  the  isometric  system 
except  the  rhombic  dodecahedron  : 
ooOoo  {100}  (P);  0,  {111}  (d); 

[•£].  .,„.,   Wi 


7r{214}  (*);    202,  {211}    (o);    and 

30,  {133}  (t).     It  is  shortened  in 

the  direction  of  one  of  the  princi- 

pal axes,  which,  together  with  the 

fact  that  only  two  of  the  three 

faces  belonging  to  the  forms  s,  o  and  t  are  developed  in 

each  octant,  gives  to  it  a  decidedly  orthorhombic  habit. 

It  has  recently  been  shown  to  be  not  improbable 
that  the  mineral  acanthite,  long  recognized  as  the 
orthorhombic  form  of  silver  sulphide,  is  only  a  distor- 
tion of  the  more  common  isometric  form  of  the  same 
substance  argentite. 

While  a  distortion  of  the  perfectly  symmetrical 
crystal  form  by  elongation  or  flattening  is  often  with- 
out any  apparent  cause,  the  same  result  is,  in  many 
other  cases,  produced  by  evident  hindrances  to  growth 
in  certain  directions.  For  instance,  such  minerals  as 

*  Neues  Jahrbuch  fur  Mineralogie,  etc.,  1889,  n.  p.  260. 
f  A.  Knop  :  Molecularconstitution  und  Wacbsthum  der  Krystalle 
(1867,  p.  50). 


208  CRYSTALLOGRAPHY. 

garnet,  tourmaline  or  quartz  diverge  from  their  usual 
habit  and  crystallize  in  the  thinnest  possible  plates 
when  they  are  formed  in  mica,  whose  perfect  cleavage 
allows  of  their  development  most  readily  along  one 
plane. 

Other  imperfections  of  crystal  form  are  due  to  the 
action  of  external  mechanical  forces,  which  cause 
bending  or  breaking.  Ductile  substances,  like  the 
metals,  or  soft  substances,  like  gypsum  or  stibnite,  are 
peculiarly  liable  to  the  first  of  these  distortions ; 
although  such  brittle  minerals  as  apatite  or  quartz, 
when  imbedded  in  crystalline  limestone,  are  frequently 
found  to  be  bent  without  breaking.  In  many  other 
cases  the  crystal  is  broken  and  its  fragments  more  or 
less  displaced  by  movements  in  its  matrix.  Crystals 
imbedded  in  granite  veins,  like  tourmaline  or  beryl, 
frequently  exhibit  this  phenomenon. 

The  most  common  imperfections  in  crystal  form  are 
due  to  the  manner  of  their  attachment  to  the  surface 
on  which  they  rest.  One  end  of  an  individual  is 
usually  prevented  in  this  way'-fey  assuming  its  charac- 
teristic planes.  Crystals  that  are  bounded  on  all  sides 
by  their  own  faces  are  of  comparatively  rare  occurrence. 

Imperfections  of  Crystal  Planes.  Theoretically  even, 
perfectly  reflecting  crystal  planes  are  almost  as  much 
the  exception  as  the  ideally  developed  crystal  forms. 
Irregularities  may  be  produced  on  crystal  planes  by 
(1)  striation,  (2)  curvature,  (3)  uneven  growth,  (4)  cor- 
rosion. 

(1)  Striation.  A  parallel  striation  of  a  crystal  face  may 
be  produced  by  the  union  of  many  individuals  in  either 
parallel  or  reversed  position.  The  first  produces 
what  was  described  in  the  last  chapter  (p.  183)  as  an 


IMPERFECTIONS  OF  CRYSTALS. 


209 


FIG.  346. 
(Pyrite.) 


oscillatory  combination  of  two  contiguous  planes,  which 

are  alternately  developed.     The  horizontal  striation, 

so  common  on  the  prismatic  faces  of  quartz  crystals, 

is  due  to  the  alternate  development  of  the  prism  and 

a  steep  rhombohedron.     Striation 

of  this   kind  must  always  accord 

with  the  symmetry  of  the  crystal, 

and  often  shows   the  hemihedral 

nature    of   a   crystal  upon  which 

no  real  hemihedral  face  appears. 

Cubes   of  iron  pyrites  frequently 

show  a  striation   of    their  planes 

in  one  direction,  which  is  perpendicular  to  the  striation 

on  all  contiguous  faces,owing  to  oscillatory  combination 

with  the  pentagonal  dodecahedron   (Fig.  346).     This 

would  not  be  possible  in  a  holohedral  cube. 

Striation  of  crystal  planes  may  also  be  produced  by 
repeated  poly  synthetic  twinning  (p.  188).  This  is  well 
illustrated  in  the  case  of  the  triclinic 
feldspar,  albite  (Fig.  347).  The  twin- 
ning plane  is  here  the  brachypinacoid, 
and  the  contact  of  a  large  number  of 
fine  lamellae,  alternately  in  twinning 
position,  would  evidently  produce  a 
striation  on  the  basal  plane  parallel  to 
the  brachydiagonal  axis.  Striations 
due  to  this  cause  are  common  on  crys- 
tals of  pyroxene,  calcite,  sphene,  aragonite,  and  many 
other  minerals.  The  direction  of  the  striation  of  course 
depends  in  each  case  on  what  face  acts  as  twinning 
plane.  In  some  instances  two  or  more  sets  of  paral- 
lel lamellae  are  intercalated  parallel  to  different  crystal 
faces  which  may  or  may  not  belong  to  the  same  form ; 


FIG.  347. 
(Albite.) 


210 


CR  7STALLOGRAPHY. 


that  is,  they  may  be  produced  by  the  same,  or  by  dif- 
ferent twinning  laws. 

(2)  Curvature  of  Crystal  Planes.  This  is  quite  a  con- 
stant property  of  some  substances.  It  may  be  due  to  a 
very  fine  oscillatory  combination  or  to  an  irregularity 
of  growth  and  a  want  of  perfect  parallelism  between 
sub-individuals.  A  common  instance  is  the  diamond, 
whose  faces  except  those  of  the  octahedron,  are  almost 
always  curved  (Fig.  348).  Crystals  of  calcite  and  gyp- 
sum are  often  curved  (Fig.  349),  while  rhombohedrons 
of  magnesian  calcium  carbonate  (dolomite)  are  some- 
times distorted  into  saddle-like  forms  (Fig.  350).  The 


FIG.  348. 
(Diamond.) 


FIG.  349. 
(Gypsum.) 


FIG.  350. 
(Dolomite.) 

twisted  quartz  crystals  of  Switzerland  are  also  instances 
of  this  kind. 

(3)  Irregularities  of  growth  often  produce  uneven- 
ness  of  crystal  planes.  Many  faces  break  up,  especially 
near  their  combination-edges,  into  other  planes  having 
very  nearly,  but  not  quite,  their  own 
position.  These  are  called  vicinal 
planes  (p.  26).  Sometimes  these  vici- 
nal planes  are  developed  in  the  form 
of  very  flat  pyramidal  protuberances 
with  more  or  less  curved  edges  and 
faces.  These-  are  very  common  on 
the  rhombohedral  faces  of  quartz  (Fig. 
351),  but  occur  to  a  greater  or  less 
extent  on  the  crystals  of  most  other  substances. 


FIG.  351. 
(Quartz.) 


IMPERFECTIONS  OF  CRYSTALS. 


211 


FIG.  352. 
(Galena.) 


In  other  cases,  uneven  planes  are  due  to  incomplete 
growth.  Many  crystals  whose 
growth  is  rapid  tend  to  form 
skeletons,  by  arranging  sub-indi- 
viduals along  the  axes  and  edges, 
and  sometimes  these  skeletons  do 
not  become  entirely  filled  up.  A 
common  instance  of  this  is  seen  in 
the  hopper-shaped  faces  of  salt 
(NaCl)  crystals,  and  in  the  artificial  crystals  of  lead 
sulphide  (Fig.  352). 

A  drusy  appearance  of  crystal  planes  is  produced  by 
the  projection  of  sub-individuals  above  the  average 
surface,  as  is  often  seen  in  the  case  of  fluorspar. 

(4)  Corrosion  subsequent  to  the  formation  of  crystals 
may  produce  irregularities  of  their  surfaces.  If  this 
action  is  extreme,  the  faces  usually  succumb  before  the 
edges  and  angles,  so  that  skeleton  forms,  closely  re- 
sembling growth  forms,  may  result.  If  the  solvent 
action  is  less,  natural  etched  figures  are  produced, 
which  well  display  the  crystals'  true  symmetry. 

A  rounding-off  of  edges 
and  angles  and  the  produc- 
tion of  a  "glazed"  appear- 
ance, as  though  there  had 
been  a  partial  fusion,  is  very 
common  on  all  crystals 
which  occur  in  crystalline 
limestone,  as,  for  instance, 
quartz,  calcite,  pyrite,  galena, 
apatite,  chondrodite,  feld- 
spar, pyroxene,  hornblende, 
tourmaline,  scapolite,  zircon,  sphene,  and  spinel. 


FIG.  353. 
(Olivine.) 


212  CRYSTALLOGRAPHY. 

This  is  doubtless  the  result  of  some  chemical  action, 
although  it  has  never  been  satisfactorily  explained. 
In  some  cases,  the  quartz  crystals  of  Herkimer  Co., 
N.  Y.,  have  their  edges  so  worn  that  only  minute 
rounded  areas  of  their  crystal  planes  remain  ;  and  the 
preceding  figure  (353)  shows  a  similar  development 
of  an  olivine  crystal  observed  by  Kose,  where  the 
planes  of  a  single  zone  are  developed  as  circular 
surfaces. 

All  imperfections  of  crystal  surfaces  express  the 
symmetry  of  the  form.  On  combinations,  some  faces 
are  dull,  while  others  are  bright ;  some  are  striated, 
while  others  are  smooth  ;  some  are  uneven,  while 
others  are  even  ;  but  all  crystallographically  equivalent 
planes  are  similarly  affected. 

False  Planes.  Apparent  crystal  faces,  whose  position 
is  not  that  of  true  crystal  planes,  may  be  produced  by 
oscillatory  combination,  as  in  the  case  of  tapering 
quartz  crystals  ;  or  by  contact,  during  crystal  growth, 
with  some  smooth  surface,  as  in  the  case  of  the  so- 
called  "  Babel  quartz." 

Variation  in  Crystal  Angles.  Most  distortions  of  form 
do  not  at  all  affect  the  interfacial  angles.  But  even 
these  are  in  some  cases  observed  to  vary,  though  gen- 
erally only  within  narrow  limits.  Such  differences  in 
angles  exhibited  by  crystals  of  the  same  substance  are 
to  be  accounted  for  (1)  by  slight  differences  in  chemi- 
cal composition  ;  (2)  by  variations  of  temperature  ; 
(3)  by  mechanical  action,  either  during  or  subsequent 
to  the  crystal's  formation  ;  (4)  by  change  of  molecular 
arrangement  through  paramorphism  or  pseudomorph- 
ism. Some  species  appear  to  be  much  more  sensitive 


IMPERFECTIONS  OF  CRYSTALS.  213 

to  these  agencies  than  others,  and  therefore  more  fre- 
quently exhibit  variations  in  their  interfacial  angles. 

Internal  Impurities  of  Crystals.  These  may  consist  of 
(1)  intermolecular  substance  in  the  form  of  a  dilute 
pigment ;  (2)  gas  inclusions  ;  (3)  fluid  inclusions  ;  (4) 
glass  inclusions  ;  (5)  inclusions  of  unindividualized 
matter  ;  (6)  crystals  of  other  substances.  Such  inclu- 
sions may  also  be  (1)  regularly  or  (2)  irregularly  ar- 
ranged ;  and  (1)  of  primary  or  (2)  of  secondary  origin. 

(1)  Dilute   pigment.    Many   crystals    are   variously 
colored    by   minute   quantities   of    matter    scattered 
molecularly  through  them.     The  nature  of  this  pig- 
ment is    generally  indeterminable.     Kemarkable  ex- 
amples are  presented  by  crystals  of  corundum  and 
tourmaline.     The  color  may  often  be  affected  by  tem- 
perature.    The  yellow  Brazilian  topaz  may  be  made 
permanently  pink  by  heating  ;  while  the  green  micro- 
cline  (Amazon  stone)    and  the  smoky  quartz  (cairn- 
gorm) may  be  decolorized  by  the  same  means. 

(2)  Gas  inclusions.  These  are  common  in  crystals  of 
both   aqueous   and   igneous   origin.     Rose   found   in 
crystals  of  rock-salt  inclusions  of  marsh-gas  and  hy- 
drogen.     The    gas   imprisoned   in    the    rock-salt  of 
Wieliczka  expands  violently  on  heating  (Knistersalz). 
Carbon  dioxide  occurs  in  the  quartz  of  many  granites 
and  other  eruptive  rocks  ;  while  traces  of  hydrocar- 
bons, sulphur  dioxide,  oxygen,  and  nitrogen  have  also 
been  noticed. 

(3)  Fluid  cavities  occur  in  crystals  of  topaz,  corun- 
dum, beryl,  diamond,  and  especially  in  quartz.     The 
shape  of  the  cavity  is  generally  irregular,  but  in  some 
cases  it  is  the  same  as  that  of  the  host  (negative  crys- 
tal).    The  fluid  is  most  commonly  water  or  an  aqueous 


214  CRYSTALLOGRAPHY. 

solution  ;  sometimes  it  is  liquid  carbon  dioxide.  A 
bubble  of  air  or  of  some  gas  is  generally  present  which 
is  frequently  movable.  Small  crystals  also  float  in 
some  of  the  liquids,  whose  size  increases  and  dimin- 
ishes with  changes  of  temperature.  Crystals  which 
grow  rapidly  from  aqueous  solutions  very  frequently 
imprison  portions  of  their  mother-liquor. 

(4)  Glass  inclusions  occur  in  crystals  formed  from 
a  molten  mass.     They  are  inclusions  of  the  mother- 
liquor  which  have  solidified  in  an  amorphous  state. 
They  also  often  contain  one  or  more  bubbles,  sur- 
rounded by  wide  black  rims,  and  of  course  immovable. 
The  presence  of  more  than  one  bubble  in  one  cavity 
is  indicative  of  a  glass  incision. 

(5)  Inclusions  of  unindividualized  matter  are  for  the 
most  part  quite  irregular,  like  clay  in  rock-salt,  gyp- 
sum and  quartz  ;  bituminous  substances  in  andalusite 
and  quartz  ;  iron  hydroxide  in  mica  and  gypsum.* 

(6)  Minute  crystals  of  various  kinds  are  often  in- 
cluded in  larger  individuals.     These   may  impart  a 
peculiar  color  or  lustre  to  the  latter,  as  in  the  case  of 
the  hematite  plates  in  carnallite  and  oligoclase  (sun- 
stone).     The  microscope  has  done  much  to  extend  our 
knowledge  of  such  associations,  which  are  very  mani- 
fold.    The  iridescence  of   minerals  like  labradorite, 
hypersthene  and  bronzite  is  due  to  minute  inclusions 
of  other  minerals  arranged  in  certain  crystallographic 
planes.     These  are  regarded  by  Professor  Judd  as  of 
secondary    origin,   occupying    cavities    produced    in 
planes  of  easiest  solution  by  percolating  waters  at  con- 


*  See  Blum,  Leonhard,  Seyffert,  and  Sochting:  Die  Einscliliisse 
von  Mineralien,     Haarlem,  1854. 


IMPERFECTIONS  OF  CRYSTALS.  215 

siderable  depths.  Professor  Judd  lias  termed  this 
process  schillerization. 

In  other  cases  quite  similar  results  are  produced  by 
original  inclusions,  as  in  the  case  of  the  sanidine  of 
some  of  the  recent  rhyolites,  and  the  zonally  arranged 
microlites  of  hauyne  and  many  of  the  Mi  Somma 
minerals. 

In  still  other  cases,  minute  mineral  impurities  may 
be  developed  by  the  incipient  alteration  of  their  host. 
By  this  means  many  crystals  of  feldspar  contain  flakes 
of  muscovite  or  kaolin,  as  well  as  needles  of  zeolites, 
scapolite,  or  zoisite.  This  subject  is  very  extensive, 
and  leads  directly  into  the  field  of  pseudomorphism, 
metamorphism  and  chemical  geology,  which  lie  with- 
out the  scope  of  the  present  work. 


APPENDIX. 


ON  ZONES,  PROJECTION  AND  THE  CONSTRUCTION  OF 
CRYSTAL  FIGURES. 

IT  is  impossible  to  include  within  the  limited  space 
of  the  present  work  the  explanations  and  formulae 
necessary  for  the  mathematical  calculation  of  crystal- 
lographic  constants  and  symbols  from  observed  inter- 
facial  angles.  The  simple  zonal  relations  existing 
between  crystal  planes  and  their  graphic  representa- 
tion by  means  of  projections  are,  however,  easily  com- 
prehended and  are  therefore  useful,  even  to  the  begin- 
ner. While  these  subjects  are  not  absolutely  essential 
to  the  understanding  of  what  has  been  given  in  the 
body  of  this  book,  they  may  nevertheless  be  employed 
with  advantage  in  connection  with  the  study  of  each 
system.  Their  consideration  is  therefore  embodied  in 
an  appendix  which  may  be  employed  as  desired. 

ZONES. 

Definition.  A  study  of  the  relation  existing  between 
the  planes  occurring  on  crystals  is  much  facilitated  by 
the  fact  that  they,  are  frequently  arranged  in  belts, 
which  extend  around  the  crystal  in  different  direc- 
tions. Such  belts  of  planes  are  technically  called 


218 


APPENDIX. 


zones.  All  planes  belonging  to  the  same  zone  are  said 
to  be  tautozonal.  The  intersection-edges  of  all  tauto- 
zonal  planes  are  parallel  to  each  other ;  and,  like  the 
planes  themselves,  are  also  parallel  to  an  imaginary 
line  passing  through  the  centre  of  the  crystal,  called 
the  zonal  axis.  The  positions  or  symbols  of  any  two 
planes  belonging  to  a  zone  are  sufficient  to  determine 
the  direction  of  its  zonal  axis.  The  symbol  of  any 
crystal  plane  is  known  if  it  be  found  to  lie  at  the 
same  time  in  two  zones  the  directions  of  whose  zonal 
axes  are  known. 

The  real  nature  of  a  zone  may  be  made  clearer  by 
an  example.  In  Fig.  354  the  planes  dy  b,  g,  c,  are  tau- 
tozonal ;  likewise  the  planes  d,  f,  h  • 
e,  b,f,  a,  ;  i,  f,  g  ;  etc.  Here  we  have 
several  distinct  zones  existing  on  the 
same  crystal ;  while  the  plane  b  be- 
longs equally  to  the  two  zones  d  b  g  c 
and  e  bf  a. 

FIG.  354.  in  Fig.  355  the  planes  oao'  form  a 

zone ;  and  it  is  equally  evident  that  the  planes  add' 
also  form  one,  in  spite  of  the  fact 
that  d  and  d'  do  not  actually  in- 
tersect. If  these  two  planes 
were  extended  until  they  did  in- 
tersect, their  edge  would  be  par- 
allel to  that  between  a  and  d. 

The  practical  determination  of 

what  planes  belong  to  the  same  zone  is  accomplished, 
so  far  as  possible,  by  the  parallelism  of  their  inter- 
secting edges.  In  cases  where  there  is  any  doubt  as 
to  the  exact  parallelism,  or  where  the  planes  in  ques- 
tion do  not  intersect,  recourse  is  had  to  the  reflecting 


APPENDIX.  219 

goniometer  (p.  22).  It  will  be  readily  understood 
that  if  a  zonal  axis  be  made  coincident  with  the  axis 
of  revolution  of  such  an  instrument,  then  each  face  of 
the  zone  must  in  turn  yield  a  reflection  as  the  crystal 
is  revolved  through  360°.  This  is  the  most  common 
and  ready  method  of  identifying  the  planes  which 
compose  a  crystal  zone. 

General  Expression  for  the  Indices  of  a  Zone.  The  es- 
sential feature  of  any  zone  is  the  direction  of  its  inter- 
section-edges, or  of  its  zonal  axis.  The  indices  of  this 
direction  may  be  obtained  from  the  indices  of  any  two 
planes  lying  in  the  zone,  and  they  are  called  the  indices 
of  the  zone. 

It  is  capable  of  geometrical  demonstration  that,  if 
the  indices  of  any  two  planes  belonging  to  a  zone  be 
indicated  by  the  letters  hkl  and  h'k'l',  then  the  indices 
of  their  zonal  axis,  which  are  usually  designated  by 
the  letters  u,  v,  w,  are  equal  to 

(W  _  Jc'l)a,    (fhf  -  lfh)b,    and    (hkf  -  h'tyc* 

The  zonal  indices  may  be  derived  from  the  indices 
of  any  two  planes  in  the  zone  by  the  following  easily 
remembered  process :  The  indices  of  the  first  plane 
are  written  twice  in  their  usual  order,  and  those  of  the 
second  plane  are  placed  directly  under  them  in  the  same 
order.  The  first  and  last  terms  are  then  cut  off.  The 
product  of  the  first  upper  and  second  lower  indices  has 
then  subtracted  from  it  the  product  of  the  second 


*  The  complete  proof  of  this,  which  is  very  simple,  is  too  long  to 
be  given  here.  It  will  be  found  in  Groth's  Physikalische  Krystallo- 
graphie,  3d  ed.,  p.  200  (1885),  and  in  Bauerman's  Systematic  Miner- 
alogy, p.  24  (1884). 


220 


APPENDIX. 


upper  and  first  lower  indices.     This  gives  u.     v  and  w 
are  similarly  obtained,  thus : 


h 

k 

I 

h 

k 

I 

X 

X 

X 

hf 

k' 

V 

h' 

V 

V 

(klf  -  kfl)  =  u'f  (W  —  I'h)  =  v ;  (hkf  -  h'k)  =  w. 

It  is  very  necessary,  when  using  this  formula,  to  pay 
particular  attention  to  the  signs. 

We  may  illustrate  this  by  the  following  concrete 
example :  Suppose  the  indices  of  two  planes  are  Oil 
and  211,  then  the  indices  of  their  intersection  or  zonal 
axis  will  be  022.  These  are  obtained  as  follows : 


0 

1101 

1 

XXX 

2 

1121 

1 

(1  X  1)  -  (1  X  1)  =  -  1  -  (-  1)  =  0  =  u  ; 
(1X2)  -(0X1)  =  -2-  0  =2  =  v, 
(Oxl) -(1X2)=  0-  2  =2=10. 

The  algebraic  sum,  remainder,  and  product  must  in 
every  case  be  used,  or  an  erroneous  result  will  be 
obtained. 

Zone  Control.  The  indices  of  any  plane  which  be- 
longs to  a  zone  whose  zonal  indices  are  known,  must, 
when  multiplied  with  the  latter,  give  an  algebraic  sum 
equal  to  zero.  If  p,  q,  r,  are  the  indices  of  the  plane 
in  question,  then 

UP  +  V(I  +  wr  —  0-* 

*  The  proof  of  this  equation  is  again  too  long  to  be  quoted  here. 
It  will  be  found  in  Groth's  Physikalische  Krystallographie,  3d  ed., 
p.  204  (1885). 


APPENDIX.  221 

This  is  called  the  zonal  equation,  and  is  used  as  a  con- 
trol in  deciding  whether  a  plane  belongs  to  a  given 
zone  or  not.  We  may  apply  it  to  test  the  result  of 
our  last  example.  If  the  planes  Oil  and  211  really 
belong  to  the  zone  whose  indices  are  022,  then 

(0  x  0)  4-  (2  x  I)  +  (2  x  1)  =  0 

and 

(0  X  2)  +  (2  X  1)  +  (2  X  1)  =  0. 

The  zonal  equation  may  also  be  sometimes  used  to 
determine  the  symbol  of  a  plane  which  truncates  an 
edge  between  two  known  planes  and  therefore  lies  in 
a  zone  with  them.  For  example,  the  edges  of  the 
rhombic  dodecahedron,  oo  0,  are  truncated  by  the  faces 
of  a  form  mOm  (Plate  II.,  Fig.  18) ;  what  is  the  value 
of  m  ?  The  edge  between  the  planes  whose  indices 
are  101  and  Oil  is  replaced  by  a  face  whose  intercepts 
are  ma  :  ma  :  a,  and  whose  indices  are  consequently 
1, 1,  m.  The  indices  of  the  zonal  axis  of  the  two  known 
planes  are  found,  as  explained  in  the  last  paragraph, 
to  be  111 ;  hence 

1  +  1  +  (1  X  m)  =  0 ;        and        m  =  2. 

The  Integrity  of  Zones.  From  the  last  section  it  will 
be  seen  that  the  existence  of  zones  is  wholly  depend- 
ent on  the  indices  of  the  planes  which  compose  them, 
and  quite  independent  of  the  relative  lengths  of  the 
axes  to  which  these  are  referred.  These  latter  values 
are  found  to  vary  slightly  with  the  temperature,  since 
the  expansion  of  a  crystal  by  heat  is  unequal  in  dif- 
ferent directions.  The  indices,  however,  retain  their 
rational  values  for  all  temperatures,  and  therefore  no 
change  of  external  conditions  can  affect  the  so-called 
integrity  of  the  zones. 


222  APPENDIX. 

Determination  of  the  Indices  of  an  Unknown  Plane  be- 
longing to  Two  Known  Zones.  Any  plane  which  is 
parallel  to  two  different  and  known  directions  has  its 
position  thereby  determined.  A  crystal  plane  which 
lies  simultaneously  in  two  zones  must  be  parallel  to 
the  zonal  axes  of  both.  If  the  indices  of  two  zones 
are  u,  v,  iv,  and  u't  v',  w',  and  the  indices  of  a  plane 
belonging  to  both,  p,  q,  r,  then 

up-\-  vq  -\-  wr  =  0 
and 

u'p  +  vfq  -|-  w'r  =  0. 

From  these  two  zone- control  equations  we  obtain 


UV   —  U  V 

and 

wu'  —  w'u 
q  =  r . 7—7 . 

UV  —  UV 

Inasmuch  as  it  is  possible  to  make  one  of  the  indices 
equal  to  any  number  without  destroying  their  relative 
values,  we  may  make  r  =  uv'  —  u'v  and  thus  obtain 

p  =  vwr  —  v'w  ; 
q  =  wu'  —  w'u ; 
r  =  uv'  —  u'v. 

These  are  the  values  of  the  indices  of  the  plane  ex- 
pressed in  terms  of  the  zonal  indices. 

The  indices  of  an  unknown  plane,  found  to  lie  simul- 
taneously in  two  known  zones,  may  therefore  be  found 
by  combining  the  zonal  indices  according  to  the  same 


APPENDIX. 


223 


method  as  was  above  given  for  finding  the  zonal  in- 
dices from  the  indices  of  two  known  planes.     Thus  : 


u 

V          W          U          V 

W 

XXX 

*' 

vf      wf      u'      v' 

11}' 

vwf  —  vfiv  =  p ;  wuf  —  w'u  =  q ;  uvf  —  u'v  =  r. 

Example.  Suppose,  on  a  certain  crystal,  an  unknown 
plane  is  found  to  lie  in  one  zone  with  two  other  planes 
whose  indices  are  110  and  101 ;  and  at  the  same  time 
to  belong  to  a  second  zone  two  of  whose  planes  have 
the  symbols  111  and  100.  What  are  the  indices  of 
the  plane  in  question  ? 


1011 

XXX 
0110 


1111 

XXX 
0010 


(1-0)  (0-1)  (0-1) 
(111),  indices  ofjlst  zone 

1 


llll 

XXX 
1101 


(0-0)  (1-0)  (0-1) 
(Oil),  indices  of  2d  zone. 

1 


(1-1)  (0-1)  (1-0) 
(211),  indices  of  plane  required. 

PROJECTION, 

Definition.  The  zonal  relations  of  crystal  planes 
are  advantageously  represented  by  graphic  methods. 
Two  such  methods  are  at  present  extensively  employed, 
both  of  them  being  systems  of  projection  proposed  by 
Neumann  in  1823. 

The  first  is  called  spherical  projection,  and  represents 


224  APPENDIX. 

the  position  of  each  plane  on  the  upper  half  of  a  crys- 
tal by  the  point  of  intersection  of  a  normal  to  the 
plane  with  the  surface  of  a  sphere  at  whose  centre  the 
crystal  is  supposed  to  be.  The  hemisphere  thus  ob- 
tained is  then  projected  on  a  plane  passed  through  its 
equator,  while  the  eye  is  imagined  at  the  opposite  ex- 
tremity of  its  polar  diameter. 

The  second  method  is  called  the  linear  projection.  It 
represents  each  crystal  face  by  its  line  of  intersection 
with  an  imaginary  plane,  called  the  plane  of  projection. 
While  neither  so  elegant  nor  so  useful  for  purposes  of 
calculation  as  the  first  method,  this  projection  possesses 
certain  peculiar  advantages  for  beginners,  and  is  the 
basis  upon  which  crystal  drawings  are  constructed. 
For  these  reasons  it  will  be  first  considered.* 

Construction  of  Linear  Projections.  If  we  imagine  all 
the  faces  of  a  crystal  to  be  shifted  without  changing 
their  direction,  until  they  cut  the  vertical  axis  at  unit 
distance  from  the  centre,  then  their  linear  projection 
is  formed  by  the  lines  of  their  intersections  with  an 
imaginary  plane  (plane  of  projection)  which  passes 
through  the  lateral  axes.  Any  plane  may  be  made  the 
plane  of  projection,  but  the  one  here  mentioned  is 
generally  selected  for  this  purpose.  The  lines  of  inter- 
section between  the  crystal  faces  and  the  plane  of  pro- 
jection are  called  the  section-lines  of  the  planes.  Every 
section-line  stands  for  a  pair  of  planes  on  all  holohe- 
dral  and  parallel-face  hemihedral  forms  ;  but  projec- 

*  Those  desiring  full  information  regarding  the  construction  and 
use  of  spherical  projections  will  find  it  in  the  works  of  Miller, 
Reusch,  Liebisch,  Groth,  and  Henrich,  cited  at  the  beginning  of  this 
book.  The  works  of  Queudstedt,  Klein,  and  Websky  contain  similar 
details  in  reference  to  linear  projections. 


APPENDIX. 


225 


tions  of  inclined-face  forms  do  not  differ  from  those  of 
the  corresponding  holohedrons,  because  the  section- 
lines  here  stand  only  for  single  faces. 

The  construction  may  be  advantageously  illustrated 
by  a  few  simple  examples. 


\ 


FIG.  356.  FIG.  357. 

The  Octahedron.  The  planes  of  this  form  already 
cut  the  vertical  axis  at  unity,  and  hence  require  no 
shifting.  The  intersection  of  the  four  upper  planes 
with  the  horizontal  axial  plane  would  evidently  give 
the  projection  in  Fig.  357. 

The  Cube.  The  upper  surface  of  the  cube  evidently 
cannot  appear  in  the  projection  since  it  cuts  the  ver- 


K 


FIG.  358. 


FIG.  359. 


tical  axis  at  unity,  but  intersects  the  plane  of  projec- 
tion only  at  infinity  (i.e.  is  parallel  to  it).     The  other 


226  APPENDIX. 

planes,  in  order  to  be  made  to  cut  the  vertical  axis  at 
unity,  must  be  shifted  until  they  include  it  through- 
out. Each  plane  will  then  coincide  with  its  parallel 
plane,  and  the  two  pairs  will  intersect  the  plane  of 
projection  in  a  rectangular  cross,  Fig.  359. 

The  most  simple  rule  for  constructing  a  linear  pro- 
jection is  to  lay  off  on  paper  (the  plane  of  projection) 
the  two  (or  three)  lateral  axes  which  lie  in  this  plane. 
Then  reduce  the  Weiss  symbol  for  each  plane  in  the 
upper  half  of  the  crystal  to  a  form  in  which  its  vertical 
axis  is  equal  to  unity.  [This  is,  of  course,  accom- 
plished by  dividing  each  term  of  the  symbol  by  the 
parameter  of  the  vertical  axis.]  When  the  symbols 
have  all  been  reduced  to  this  form,  the  section-line 
for  any  plane  may  be  obtained  by  merely  connecting 
those  points  on  the  lateral  axes,  whose  positions  are 
indicated  by  the  relative  values  of  the  new  lateral 
parameters. 

Examples.  The  form  shown  in  Fig.  360  is  the  iso- 
metric icositetrahedron  (p.  55),  composed  of  three 
planes  in  each  octant  whose  symbols,  according  to 
Weiss's  notation,  are  : 


These   symbols,  when  reduced  to  a    form  in  which 
c  =  1,  become  : 

(  2a  :  2a  :  a  } 

1    a  :  £a  :  a  v. 
(  %a  :    a  :  a  ) 

From  these  values  the  projection  shown  in  Fig.  361 
is  readily  constructed. 


APPENDIX. 


227 


+a 
FIG.  360.  FIG.  361. 

The  next  example  represents  a  rhombohedral  com- 
bination occurring  on  the  mineral 
tourmaline  (Fig.  362).  Its  eight  forms 
are  as  follows : 

oo  P,  11010}  (1) ;    oo  P2,  j  1120 1  (s) ; 
R,  *{10ll|  (j>);    -fR,  ^{0112}  W 
-25,  /f  1 0221  ( (o);  53,  *  {3121}  (Q ; 
55,  /cf5231J  (u) ;     -  2^2,  ^S^i?  (v) 

The  zonal  relations  of  thelse  planes 
are  manifold,  and  may  be  exhibited 
with  great  distinctness  upon  a  linear  projection  like 
that  shown  in  Fig.  363. 

The  third  example  shows  a  triclinic  crystal  whose 
planes  are  referred  to  axes  of  unequal  length  and  ob- 
liquely inclined  to  each  other.  The  two  lateral  axes 
intersect  at  an  angle  of  131°  33r.  The  Weiss  symbols 
of  the  planes  are  as  follows  : 


FIG.  362. 


228 


P  =  a  :  —  biccc 
u  =  a  :      b  :  oo  c 

Axial  ratio :    a  :  b  =  0.492  +  : 


r  =  a 
x  =  a 

s  =  a 


-b:   c 
b:    c 

co  b  :  2c 


FIG.  365. 


FIG.  364. 

A  careful  consideration  of  these  examples,  together 
with  what  has  been  said  before  with  reference  to  the 
linear  projection,  will  make  clear  the  truth  of  the  fol- 
lowing important  points : 


APPENDIX.  229 

1.  After  the  supposed  shifting  of  the  planes  to  a 
position  where  they  all  cut  the  vertical  axis  at  unity, 
all  tautozonal  planes  will  intersect  in  a  single  line 
which  is  the  direction  of  their  zonal  axis.     But  as  a 
rule  this  line  intersects   the   plane   of   projection   in 
a  pointy  through  which   all  the  section-lines   of  the 
planes  belonging  to  this  zone  must  pass.     Such  a  point 
on  the  projection  is  called  a  zone-point. 

2.  If  the  direction  of  any  zonal  axis  is  parallel  to 
the  plane  of  projection,  then  all  the  section-lines  of 
this  zone  will  be  parallel,  i.e.  will  intersect  at  infinity. 

3.  All  planes  which  are  parallel  to  the  vertical  axis 
must    be    represented   by   section-lines   which    pass 
through  the  central  point  of  the  projection.     This  is 
because  the  zonal  axis  for  all  such  planes  is  the  ver- 
tical axis.     The  direction  of  such  lines  is  determined 
by  the  relative  value  of  their  intercepts  on  the  lateral 
axes.* 

Symbol  of  any  Plane  belonging  to  Two  Zones  obtained 
by  Linear  Projection.  The  linear  projection  presents 
another  ready  means  of  obtaining  the  symbol  of  any 
plane  lying  simultaneously  in  two  zones.  The  inter- 
section of  any  two  section-lines  in  the  projection  is 
enough  to  fix  the  position  of  the  zone-point  of  the  zone 
to  which  they  belong.  If  the  two  zone-points  of  the 
zones  in  which  the  unknown  plane  lies  can  be  deter- 
mined in  this  way,  its  section-line  in  the  projection  is' 
found  by  merely  connecting  them.  From  the  section- 


*  The  construction  of  linear  projections  of  various  crystal  forms 
should  be  made  a  matter  of  constant  practice  by  the  student  until 
the  subject  presents  no  difficulty,  and  until  he  can  appreciate  the 
full  significance  of  such  projections  at  a  glance. 


230 


APPENDIX. 


a:  oo  o:  <»c 


line  thus  obtained  the  symbol  of  the  plane  can  readily 

be  deduced. 

Example.     Suppose  a  plane  lies  in  a  zone  with  the 

planes  a  :  b  :  oo  c  and 
a  :  GO  b  :  c,  and  at  the 
same  time  in  a  second 
zone  with  the  planes 
/  a:b:c  and  a  :  oo  b  :  oo  c ; 
what  is  its  symbol  ? 

The  symbol  of  the 
plane  (dotted  section- 
line  in  Fig.  366)  must  be: 
%a  :  b  :  c  =  a:2b:2c. 
This  is  the  same  case 
as  that  solved  by  the 

other  method  (p.  223),  and  the  results  will  be  seen  to 

agree.* 

*  A  few  examples  for  practice  are  here  given.  After  they  have 
been  solved  by  both  of  the  above-explained  methods  other  examples 
should  be  taken  until  the  subject  is  satisfactorily  mastered. 

What  plane  lies  in  each  of  the  following  pairs  of  zones  ? 


a:  oofr.-c 
1st  Zone-ooint 


FIG.  366. 


Weiss's  Symbols. 

Naumann's 
Symbols. 

Miller's 
Symbols  (Indices). 

l.j     « 
1     a 

aoa 
2a 

oca          ooa 
2a             a 

a 

2a 

ooa 

-2a 

ooOoo  :  ooOoo 
2O2   :    2O2 

100    OlO(lstzone) 
211    211(2dzone) 

2.-jc°a 
'     a 

ooa 
ooa 

a             a 
oca             a 

2a 
a 

ooa 
a 

ooOoo 
ooOoo 

00  O2 

o 

001    210 
100    111 

S.-j00" 

I     a 

ooa 
ooa 

a             a 
ooa            2a 

2a 
a 

ooa 
2a 

ooOoo 
ooOoo 

00  O2 

202 

001    210 
100    121 

4.]     * 
1     a 

—a 
—a 

ooc             a 
2c             a 

a 
a 

c 

ooc 

OOP 

2P 

P 

OOP 

110    111 
221    110 

5.]     « 

1     a 

a 
—a 

3c             a 
ooc              a 

—a 
a 

3c 
c 

3P 

OOP 

3P 

T> 

331    331 
110    111 

6  J     a, 
1  2a, 

oo  ae 

2a2 

—  a3  :  2c   2aj 
—  as  :  2c     ax 

2a2 

oo  a2 

—  as  :  ooc 
—  a8  :  ooc 

2P 
2P2 

QO  P2 

OOP 

2021  1120 
1121  1010 

APPENDIX. 


231 


Spherical  Projection.  The  other  method  of  graphi- 
cally representing  the  relationships  of  crystal  planes 
mentioned  on  p.  224  is  called  spherical  or  stereographic 
projection.  The  principle  upon  which  such  projec- 
tions are  constructed  may  be  understood  by  reference 
to  Fig.  367.  Suppose  that  we  imagine  a  crystal — here 


FIG.  367. 

the  simple  isometric  combination  of  cube  and  rhom- 
bic dodecahedron — so  placed  that  its  center  coincides 
with  the  center  of  a  sphere,  while  its  vertical  axis  is 
also  coincident  with  the  vertical  axis  of  the  sphere. 
If  lines  be  drawn  from  the  center  of  the  sphere,  nor- 
mal to  each  of  the  crystal  planes,  they  will  intersect 
the  surface  of  the  sphere  in  points,  called  the  poles  of 
the  planes.  The  distribution  of  the  poles  A,  B,  Bf,  C, 
D,  1)',  etc.,  on  the  upper  hemisphere,  definitely  fixes 
the  relative  positions  of  the  planes  a,  b,  b',  c,  d,  df,  etc., 
on  the  crystal. 


232 


APPENDIX. 


The  arrangement  of  the  poles  on  the  upper  hemi- 
sphere is  represented  on  a  plane  surface  as  it  would 
appear  to  the  eye  situated  at  the  lower  extremity  of 
the  vertical  axis,  E;  in  other  words,  the  poles  are 
projected  on  the  equatorial  circle  of  the  sphere, 
BD"AD"fB't  which  is  called  the  fundamental  circle, 
(German,  Grundkreis).  In  this  way  the  projection  of 
any  pole  becomes  the  point  of  intersection  between  a 
line  joining  it  with  the  lower  end  of  the  meridian  axis, 
Et  and  the  equatorial  circle.  Thus  the  projection  of 
the  pole  D  is  # ;  of  D' ',  tf' ;  etc.  The  complete  spheri- 
cal projection  of  the  combination  shown  in  Fig.  367  is 
given  in  Fig.  368. 


iio 


FIG.  368. 

A  consideration  of  the  above-described  example  will 
illustrate  the  following  properties  of  spherical  pro- 
jections: 

1.  For  all  crystals  with  rectangular  axes  the  pole  of 
the  basal  pinacoid  {001}  will  occupy  the  central  point 
of  the  projection. 


APPENDIX. 


233 


2.  The  poles  of  all  planes  belonging  to  the  prismatic 
zone,  i.e.  parallel  to  the  vertical  axis,  will  lie  in  the 
circumference  of  the  fundamental  circle. 

3.  The  poles  of  all  planes  belonging  to  the  same 
zone  will  fall  in  the  circumference  of  the  same  great 
circle  (zonal  circle)-,  and  the  same  is  true  of  the  pro- 
jections of  such  poles. 

4.  All  zonal  circles  whose  axes  are  horizontal  ap- 
pear in  the  projection  as  diameters  of  the  fundamen- 
tal circle. 


1010 


0110 


m 


m 


m 


FIG.  369.  FIG.  370. 

5.  The  angular  distance  between  any  two  poles  or 
their  projections,  measured  on  their  zonal  circle,  is 
equal  to  the  normal  angle,  or  supplement  of  the  inter- 
facial  angle  included  between  the  planes  to  which  the 
poles  belong. 

The  last-named  property  of  spherical  projections 
renders  them  particularly  valuable  as  aids  in  crystal- 
lographic  calculation,  but  the  details  of  their  appli- 
cation to  this  end,  as  well  as  the  method  of  their  prac- 
tical construction,  must  be  sought  for  in  larger  works. 


234 


APPENDIX. 


The  spherical  projection  of  a  crystal  admirably  ex- 
presses its  symmetry,  as  may  be- seen  from  the  two 
following  examples.  Fig.  369  is  the  projection  of  the 
holohedral  hexagonal  combination  observed  on  beryl 
(Fig.  370)  as  described  on  page  116.  The  dots  without 
indices  are  the  projections  of  the  poles  of  the  twelve 
faces  of  the  dihexagonal  pyramid,  3Pf,  {3211)  («). 

Fig.  371  shows  by  spherical  projection  the  relation  of 
the  planes  which  commonly  occur  on  the  monoclinic 
feldspar,  orthoclase  (Fig.  372),  described  on  page  166. 


HP 


FIG.  371.  FIG.  372. 

THE  CONSTRUCTION  OF  CRYSTAL  FIGURES, 

Method  of  Representing  Crystal  Forms.  The  accurate 
construction  of  crystal  figures  is  a  matter  of  much 
importance,  particularly  in  connection  with  descrip- 
tions of  crystals  of  new  substances  or  of  such  as 
possess  unusual  or  complicated  habits.  It  is  custom- 
ary to  represent  crystals  in  their  ideal  development, 
i.e.  as  free  from  all  distortion  of  form  or  irregularity 


APPENDIX. 


235 


of  growth,  unless  these  possess  some  peculiar  signifi- 
cance. 

The  parallelism  of  crystal  edges  is  so  important,  as 
indicating  the  existence  of  zones,  that  it  is  most  desir- 
able to  retain  this  feature  in  the  .crystal  figure.  For 
this  reason  ordinary  perspective  figures  are  not  em- 
ployed, but  rather  such  as  represent  the  crystal  at  an 
infinite  distance  from  the  observer.  In  this  way  all 
rays  coming  to  the  eye  are  parallel,  and  a  projection 
is  formed  wherein  all  the  edges  belonging  to  the  same 
zone  are  parallel  in  the  figure,  as  they  are  on  the  actual 
crystal. 

Projections  like  those  here  described  are  of  two  kinds, 
according  as  the  parallel  rays  passing  from  the  crystal 
to  the  eye  are  normal  or  oblique  to  the  plane  upon 


M 


FIG.  373. 


Fia.  374. 

which  the  figure  is  projected.     The  former  are  called 
orthographic,  and  the  latter  dinographic  projections. 

Orthographic  Projections.  It  is  customary  to  con- 
struct orthographic  projections  of  crystals  upon  a  hori- 
zontal plane  (which  in  all  systems  with  rectangular 
axes  is  the  basal  pinacoid),  while  the  eye  is  conceived 
of  as  at  an  infinite  distance  in  the  direction  of  the 
vertical  axis. 


236  APPENDIX. 

Thus,  Fig.  373  represents  the  orthographic  projec- 
tion upon  the  basal  plane  of  a  complicated  crystal  of 
topaz,  whose  vertical  clinographic  projection  is  given 
in  Fig.  374.  The  forms  of  this  combination  are 
OP,  {001}  (P);  ooP,  {110}  (Jtf);ooP2,  {120}  © ; 
oo  P4,  {140}  (n);P,  {111}  (o);  fP,  {112}  (u) ;  JP, 
{113}  (t);  Pc5b,  {101}  (d);  ^Poo,  {103}  (A);  |P&, 
{023}  (a);  Pdb,  {011}  (/);  2P&,  {021}  (y) ;  4Poc, 
{ML  j(io);  and2P2,  {121}  (r). 

The  most  ready  and  convenient  way  to  construct  an 
orthographic  projection  of  any  crystal  is  to  prepare  a 
linear  projection  (p.  224)  of  all  of  its  forms  upon  the 
same  plane  as  that  selected  for  the  orthographic  pro- 
jection. If  now  the  central  point  of  this  linear  projec- 
tion is  connected  with  the  point  of  intersection  of  any 
two  section-lines,  the  direction  of  the  edge  between 
the  two  planes  corresponding  to  these  section-lines  is 
thereby  obtained  for  the  orthographic  projection. 

It  is  sometimes  desirable  to  construct  orthographic 
projections  of  monoclinic  crystals  on  the  plane  of  their 
clinopinacoid.  When  either  this  or  the  basal  pinacoid 
is  selected  as  the  plane  of  projection  the  method  of 
procedure  is  that  above  given.  If,  however,  it  is 
desired  to  construct  an  orthographic  projection  of  a 
monoclinic  crystal  upon  a  plane  perpendicular  to  the 
vertical  axis,  the  clinodiagonal  axis  should  not  then 
be  represented  at  its  full  length  as  compared  with  the 
orthodiagonal,  but  its  length  should  be  d  .  sin  /3. 

If  it  is  desired  to  construct  an  orthographic  projec- 
tion of  a  triclinic  crystal  upon  a  plane  perpendicular 
to  the  vertical  axis,  the  two  lateral  axes  should  be 
made  to  intersect  at  the  angle  included  between  its 
two  vertical  pinacoids,  while  their  lengths  should  be 
proportional  to  the  values  a  .  sin  ft  and  b  .  sin  a. 


APPENDIX.  237 

These  values  must  be  employed  in  making  the  linear 
projection,  from  which  the  orthographic  projection  is 
derived  as  above  described. 

Clinographie  Projections.  If  the  eye,  still  conceived  of 
as  at  an  infinite  distance,  is  not  directly  in  front  of  the 
crystal,  but  to  one  side,  then  the  parallel  rays  which 
reach  it  are  oblique  to  the  plane  of  projection  and  a 
clinographic  projection  results. 

It  is  customary  to  represent  crystals  by  their  clino- 
graphic projections  drawn  in  a  vertical  position  with 
the  eye  turned  a  certain  angular  distance  (6)  to  the 
right,  and  elevated  a  certain  angle  (e)  above  the  center 
of  the  crystal.  In  this  way  its  right  side  and  top  are 
brought  into  view.  Such  projections  differ  from  ordi- 
nary perspective  figures  in  having  no  vanishing  point, 
i.e.  they  show  as  parallel  all  lines,  whatever  is  their 
direction,  which  are  parallel  on  the  object.  They  are 
therefore  examples  of  parallel  perspective. 

In  order  to  construct  a  clinographic  projection  of  a 
crystal,  it  is  necessary  to  know  (1)  the  values  of  its 
crystallographic  constants,  and  (2)  the  crystallographic 
symbols  of  its  planes.  The  first  step  in  the  construc- 
tion of  the  projection  is  then  the  preparation  of  a 
perspective  view  of  the  axes,  in  exactly  the  position 
desired  for  the  finished  figure. 

It  is  usual  to  assume  such  values  for  the  angular 
revolution  and  elevation  of  the  eye  (#  and  e)  as  can  be 
expressed  by  a  simple  ratio  between  the  projected  axes, 
when  their  actual  lengths  are  equal  (isometric  system). 

The  values  of  the  angles  $  and  e  are  determined  as 
cot  d  =  r,  cot  e  =  rs,  it  being  usual  to  make  r  =  3  and 
5  =  2.  In  this  case  we  have  (Fig.  375) 

01:  OK'  ::  I  :  3    and    IA  :  10  ::  1  :  2, 
when  the  value  of  d  is  18°  26',  and  of  e,  9°  28'. 


238 


APPENDIX. 


Projection  of  the  Axes  for  the  Isometric  System.  If  the 
values  for  r  and  s  be  assumed  as  above,  then  the 
method  of  construction  of  the 
isometric  axes  is  as  follows  (Fig. 
375) :  Draw  two  lines  LLf  and 
KK'  at  right  angles  to  one  an- 
other. Make  KO  =  K'0,  and 
divide  KK'  into  three  equal 
parts.  Draw  verticals  through 
the  four  points  thus  obtained 
on  KK',  and  below  Kr  lay  off 
K'H=%K'0.  Draw  HO,  which 
will  give  the  direction  of  the 
front  lateral  axis.  Its  length 
will  be  that  portion  of  this  line  included  between  the 
two  inner  verticals,  A  and  A'. 

Draw  AS  parallel  to  K' 0  and  connect  the  points  S 
and  0.  From  the  intersection  of  this  line  with  the 
inner  vertical,  T,  draw  TB  parallel  to  K'K.  From 
point,  B,  thus  obtained  draw  the  line  BBf  through  0. 
This  will  be  the  second  lateral  axis. 

Below  K,  lay  off  KQ  =  \OK  and  make  00=  00' 
=  OQ ;  then  CCf  will  be  the  length  of  the  vertical  axis. 
Projection  of  the  Axes  for  the  Tetragonal  and  Orthorhom- 
bic  Systems.  The  axes  constructed  for  the  isometric 
system  may  be  readily  adapted  to  both  the  other  sys- 
tems with  rectangular  axes  by  merely  laying  off  por- 
tions of  the  lines  AA'  and  CC'  (Fig.  375),  which  are 
proportional  to  the  lengths  expressed  in  the  axial 
ratios  of  the  crystals  to  be  figured. 

In  the  case  of  a  tetragonal  crystal  like  zircon,  whose 
axial  ratio  is  a  :  c  ::  1  :  .64,  the  two  lateral  axes  remain 
unchanged,  while  the  vertical  axis  must  be  made  .64 
of  the  length  CC'. 


APPENDIX. 


239 


C' 
FIG.  376. 


For  an  orthorhombic  crystal  the  axis  BB'  alone  re- 
mains unchanged,  while  A  A  and  CG'  are  both  reduced 
to  the  proportionate  lengths  belonging  to  the  substance 
in  question. 

Projection  of  the  Monoclinic  Axes.  To  project  the  in- 
clination, /?,  of  the  clinodiagonal  axis,  we  construct  the 
axes  as  in  the  isometric  system, 
and  then  lay  off  Oc  =  OG .  cos  /?, 
and  on  OA  lay  off  Oa  =  OA .  sin 
ft.  From  c  draw  a  line  parallel  to 
OA',  and  from  a  another  parallel 
to  OG.  From  their  intersection,  a 
line  (DDr)  drawn  through  0  will 
give  the  direction  of  the  clino-axis 
(Fig.  376).  The  relative  lengths 
of  the  axes  must  now  be  determined,  according  to  the 
axial  ratio  of  the  substance,  as  in  the  orthorhombic 
system. 

Projection  of  the  Triclinic  Axes.  In  this  case  all  three 
axes  of  reference  intersect  obliquely  b /\c  =  ot,  a/\c  —  /?, 
a /\b  =y.  If  we  start  with  the 
isometric  axes,  the  first  step  in 
their  adaptation  to  the  triclinic 
system  is  to  obtain  the  direc- 
tion of  the  two  vertical  axial 
planes  or  pinacoids.  To  do 
this,  we  lay  off  on  OB,  Ob  = 
OB .  sin  0  (0  being  the  angle 
oo Poo,  {100}  A  oo P&,  {010}, 
which  is  evidently  not  the  same 
as  y),  and  on  OA,  Oa—OA.  cos  0  (Fig.  377).  The  line 
drawn  from  the  angle  d  of  the  parallelogram  adbO 
through  0  will  give  the  direction  of  the  macropina- 


240 


APPENDIX. 


coidal  section,  DD'.  To  obtain  the  direction  of  the  ma- 
crodiagonal  axis  (b),  lay  off  on  ODf,  Odf  =  QD*  .  sin  a  ; 
and  on  0  (7,  Oc  =  0  C  .  cos  a.  From  the  parallelo- 
gram, d'OcK',  thus  obtained,  the  diagonal,  K'K,  gives 
the  macrodiagonal  axis.  In  a  similar  manner,  the 
brachy  diagonal  axis  (a),  HHf,  is  found  by  laying  off 
on  OA',  Oa'=OA'.sin  /?  ;  and  upon  OC,  Oc'  = 
OC  .  cos  ft.  After  the  axes  have  been  projected  in 
their  proper  directions,  their  relative  lengths  must  be 
given  them  in  accordance  with  the  axial  ratio  of  the 
substance,  just  as  in  the  orthorhombic  and  monoclinic 
systems. 

Projection  of  the  Hexagonal  Axes.  These  may  be  pro- 
jected in  a  manner  analogous  to  that  given  for  the 
isometric  axes,  as  explained  by 
Dana  (see  literature  references  be- 
low). A  simpler  method  is,  how- 
ever, as  follows:  Construct  an 
orthorhombic  set  of  axes  whose 
axial  ratio,  a  :  b  :  c,  is 


=  1.732)  :  1  :  c 

(c  being  given  the  value  of  the 
vertical  axis  belonging  to  the  substance  to  be  drawn)  ; 
connect  the  extremities  of  the  two  lateral  axes,  and,  in 
the  rhomb  thus  formed,  the  obtuse  angles,  at  the  ends 
of  the  b  axis,  will  be  exactly  120°.  If  lines  be  now 
drawn  parallel  to  5,  through  points  on  the  axis,  a, 
half  way  between  its  extremities  and  the  center,  o,  the 
rhomb  will  be  converted  into  a  hexagon,  with  all  of 
its  angles  exactly  120°.  If  we  connect  the  diagonally 
opposite  angles  of  this  hexagon,  we  shall  obtain  the 
projection  of  the  hexagonal  axes  required  (Fig.  378). 


APPENDIX.  241 

Construction  of  Crystal  Figures  upon  the  Axes.  After 
the  axes  for  any  particular  substance  have  been  con- 
structed according  to  the  methods  above  explained, 
the  next  step  is  to  erect  upon  them  the  complete  clino- 
graphic  projection  of  the  crystal  whose  figure  is  desired. 
The  manner  in  which  this  is  accomplished  is  the  same 
for  all  the  systems.  Such  figures  consist  of  a  series  of 
lines  representing  in  parallel  perspective  the  combina- 
tion-edges between  the  crystal  planes.  It  is  first  neces- 
sary to  determine  the  proper  direction  of  each  of  these 
edges,  and  then  they  may  be  united  so  as  to  represent 
the  particular  combination  or  habit  desired.  For  the 
latter  purpose  it  is  desirable  to  have  as  a  guide  a  free- 
hand sketch  showing  approximately  the  relative  de- 
velopment and  distribution  of  the  various  planes. 

The  simpler  forms  of  each  system  may  be  con- 
structed directly  upon  the  projected  axes  in  a  way 
which  requires  no  particular  explanation.  For  in- 
stance, connecting  the  extremities  of  the  axes  pro- 
duces the  ground-form  of  the  substance  to  be  drawn. 
Vertical  lines  through  the  extremities  of  the  lateral 
axes  give  the  fundamental  prism,  etc. 

This  method  is,  however,  not  readily  applicable  to 
the  construction  of  figures  of  more  complex  combina- 
tions. Another  method  is  therefore  usually  employed 
for  these,  which  is  based  upon  the  use  of  the  linear 
projection.  A  complete  linear  projection  of  the  crys- 
tal to  be  drawn  is  first  prepared  in  the  ordinary  manner 
(p.  224),  and  this  is  then  thrown  into  parallel  perspec- 
tive upon  the  axes  by  connecting  points  on  the  lateral 
axes,  which  correspond  to  those  so  connected  on  the 
linear  projection.  When  this  has  been  done,  the  direc- 
tion of  any  combination-edge  is  found  by  merely  connect- 
ing the  point  ivhere  the  two  section-lines  (representing  the 


242 


APPENDIX. 


planes  forming  the  edge  in  question)  meet  wiih  the  unit 
distance  on  the  vertical  axis.  The  directions  of  the  vari- 
ous edges  thus  obtained  may  be  combined  in  the  way 
they  appear  on  the  crystal.  After  lengths  have  been 
assumed  for  one  or  two  of  the  lines,  according  to  the 
size  of  the  drawing  desired,  the  lengths  of  the  other 
lines  will  be  determined  by  their  mutual  intersections. 
This  method  of  construction  may  be  advantageously 
illustrated  by  an  example.  Required  to  project  in 

parallel   perspective  a 
crystal      of      sulphur, 
n/  showing     the     forms : 


);  andPdb, 

1 011}  (n).  The  axial 
ratio  of  this  substance 
is  0.813:1: 1.904.  Fig. 
379  gives  an  ordinary 
linear  projection  of 
these  forms,  and  Fig. 
380,  the  same  thrown  into  perspective  upon  a  set  of 


FIG.  380. 


APPENDIX.  243 

orthorhombic  axes  having  the  lengths  required  by  the 
axial  ratio  of  sulphur.  The  shape 
of  the  basal  pinacoid,  c,  in  the 
figure  is  given  by  its  intersections 
with  s,  and  it  must  therefore  be 
that  of  the  rhomb,  ss's"s"f.  The 
size  of  this  rhomb  must  be  deter- 
mined according  to  the  dimen- 
sions desired  for  the  figure. 

From  the  right-hand  corner  of 
the  face,  c,  descend  two  lines  rep-  Fm>  881< 

resenting  the  combination-edges  of  the  plane  n  with 
the  front  and  back,  right-hand  faces  of  the  obtuse 
pyramid  (s  and  «').  The  directions  of  these  two 
edges  are  obtained  by  joining  the  points  n  and 
nf  (the  intersections  of  the  section-line  of  the  plane 
n  with  those  of  the  planes  s  and  s')  with  c  ( =  1) 
(Fig.  380).  Two  lines  with  these  directions  should  be 
drawn  from  the  right-hand  corner  of  the  rhomb  c,  and 
the  length  of  one  of  them  determined  so  as  to  give  to 
the  face  s  approximately  the  relative  size  that  it  has 
on  the  crystal.  When  this  is  done,  the  length  of  the 
other  line  is  determined  by  its  intersection  with  a 
light  line  drawn  from  the  lower  extremity  of  the  first 
one,  parallel  to  the  axis  ss". 

In  the  same  manner  the  direction  of  the  combina- 
.  tion-edge  s  :  s  is  determined  by  connecting  the  point  s 
with  c ;  the  edge  p  :  n  is  obtained  by  joining  pr  and  c ; 
the  edge  p  :  p  by  connecting^  and  c ;  etc.,  etc. 

It  is  of  course  necessary  to  determine  the  direction 
of  only  one  edge  in  a  zone,  since  all  other  edges  be- 
longing to  this  zone  must  be  parallel  to  it.  The  recol- 
lection of  this  fact  will  greatly  facilitate  the  construc- 
tion. 


244  APPENDIX. 

The  lower  half  of  the  crystal  figure  may  be  con- 
structed either  by  joining  the  same  points  used  in  the 
upper  half  with  the  lower  extremity  of  the  vertical 
axis,  or  by  using  the  posterior  half  of  the  perspective 
linear  projection  in  connection  with  the  upper  ex- 
tremity of  the  vertical  axis. 

After  a  certain  amount  of  practice,  complicated  fig- 
ures can  be  rapidly  and  accurately  constructed  on  this 
principle.  To  avoid  the  confusion  of  too  many  lines 
in  the  perspective  projection,  the  axes  should  be  drawn 
in  large  proportions,  and  the  directions  of  the  edges  not 
combined  directly  upon  them,  but  on  one  side  ;  the  lines 
being  transferred  by  a  parallel  ruler,  or  by  a  ruler 
and  triangle,  to  another  part  of  the  paper.  It  is  also 
desirable  to  draw  the  finished  figure  somewhat  larger 
than  the  size  intended  for  reproduction,  and  to  have 
it  subsequently  reduced  by  photo-engraving.  If  the 
linear  projection  is  very  complex,  it  is  not  necessary 
to  transfer  it  entire  to  the  axes,  but  only  such  a 
portion  of  each  section-line  as,  by  its  intersection 
with  other  lines,  will  give  the  zone-points  required. 

Construction  of  Figures  of  Twin  Crystals.  This  de- 
pends entirely  upon  securing  two  sets  of  axes,  one  of 
which  occupies  a  position  as  though  it  had  been  re- 
volved 180°  about  a  line  normal  to  some  given  plane 
(the  twinning  plane).  Suppose  (Fig.  382)  that  abc  rep- 
resent the  relative  lengths  of  the  axes,  and  XYZ  the 
position  of  the  twinning  plane.  It  is  required  to  con- 
struct a  normal  from  0  to  the  plane  XYZ.  From  X 
draw  XL  parallel  to  ac  ;  and  from  Z,  ZH  parallel  to  ac. 
Construct  the  parallelogram  OLDH  and  draw  OD. 
In  the  same  manner  draw  YL'  and  ZK,  both  parallel 
to  be,  and  construct  the  parallelogram  OL'FK.  Draw 
OF.  If  from  the  two  points  R  and  S  straight  lines 


APPENDIX.  245 

be  drawn  to  the  opposite  angles  of  the  triangle  XYZ, 


FIG 

then  their  intersection,  P,  will  be  the  point  of  emer- 
gence of  a  normal  from  0  to  the  plane  XYZ. 

If  now  OP  be  prolonged  to  double  its  length  at  P' 
(Fig  383),  and  if  this  point  be  connected  with  X,  Fand 
Z,  then  the  lines  P'X,  P'Y, 
and  P'Z  are  the  axes  in  the 
twinning  position  required,  and 
of  lengths  corresponding  to 
the  parameters  of-  the  plane 
XYZ.  To  reduce  them  to  unit 
lengths,  corresponding  to  dbc, 
we  must  draw  aa',  W,  and  cc' 
all  parallel  to  OP. 

Upon  this  double  set  of  axes 
the  forms  belonging  to  each  individual  are  constructed 
in  the  same  manner  as  has  been  explained  for  simple 
crystals.  Each  is  drawn  entire  if  penetration  twins 


FIG.  383. 


246  APPENDIX. 

are  to  be  represented,  and  in  part  if  a  figure  of  a  con- 
tact twin  is  desired. 


For  the  convenience  of  those  desiring  fuller  informa- 
tion in  regard  to  the  construction  of  crystal  figures, 
the  following  references  are  appended  : 

C.  F.  NAUMANN  :  Lehrbuch  der  reinen  und  angewandten 

Krystallographie.     Yol.  II. 
J.  WEISBACH  :  Anleitung  zum  axonometrischen  Zeich- 

nen.     Freiberg,  1857. 
C.  KLEIN  :  Einleitung  in  die  Krystallberechnung,  pp. 

381-393.     Stuttgart,  1876. 
E.  S.  DANA:  Text-book  of  Mineralogy,  Appendix  B. 

New  York,  1883.     2d  Ed. 
TH.  LIEBISCH  :   Geometrische  Krystallographie,  Cap. 

IX.     Leipzig,  1881. 
V.  GOLDSCHMIDT  :    Ueber  Projection  und   graphische 

Krystallberechnung.     Berlin,  1887. 


INDEX. 


Acanthite,  152,  207 
Adjustment  (goniometer),  23 
Aggregates  (crystal  and  crystal- 
line), 16,  180 
Alhite,  199,  209 
Amalgam,  61 
Amorphous  substances,  7 
Andalusite,  153 
Angle  /3,  159 
Anorthite,  177 
Apatite,  121 
Aragonite,  188 
Arsenopyrite,  196 
Augite,  198 

Axes,   crystallographic,   24,  47, 
82,  105 

of  symmetry,  34 

twinning,  184 

zonal,  217 

projection  of,  238 
Axial  planes,  24,  82,  105 

ratio,  83 

Barium  nitrate,  80 
Basal  edges,  87 

pinacoid,  90,  114 
Beryl,  116,  233 
Bevelment  (of  edges),  38 
Boracite,  75 
Boron,  95 

Brachydiagonal  axis,  143,  171 
Brachydomes,  148,  174 
Brachy  pinacoid,  149,  174 
Brachyprisms,  148 
Brachypyramids,  147,  172 

Calamine,  43,  157,  187 
Calcite,  130,  194, 195 
Calcium  hyposulphite,  175 
Cane  sugar,  169 
Centering  (goniometer),  23 
Cerussite,  153,  196 
Chalcocite,  151, 197 
Chalcopyrite,  92,  102,  194 


|  Chrysoberyl,  189 
I  Clinodiagonal  axis,  159 
I  Clinodomes,  163 
I  Clinopinacoid,  163 
Clinoprisms,  163 
Clinopyramids,  162 
Clinographic  projection,  237 
Combination,  crystal,  36 

oscillatory,  183,  209 
Combinations,  isometric,  59,  69, 
74,80 

tetragonal,  93,  99,  102 

hexagonal,  114, 121,127, 136, 
140 

orthorhombic,  150 

monoclinic,  164 

triclinic,  175 

Composition  face  (twins),  185 
Compound  twins,  188 
Congruent  forms,  67  [20 

Constancy  of  interfacial  angles, 
Contact  twins,  185 
Corrosion  of  crystal  planes,  211 
Copper,  192 
Copper  sulphate,  176 
Crystal  (def.),  1 

figures,  construction  of,  241 

form  (def.),  35 
types  of,  36 

growth,  10 

habit,  11 

series,  92 

systems,  43 
Crystallizing  force,  9 
Crystallographic  axes,  24 

constants,  144,  159,  171 

notation,  27 

Crystallography  (def.),  3 
Cube,  56 

Curvature  of  crystal  planes,  210 
Cyclic  twins,  188 

Dana's  system  of  crystallographic 
notation,  29,  127 

247 


248 


INDEX. 


Diamond,  187,  210 
Didodecahedron  or  Diploid,  67 
Digonal  axes  (isometric),  47 
Dihexagonal  prism,  114 
Dihexagonal  pyramid,  109 
Diopside,  166 
Dioptase,  140 
Disthene,  200 
Distortion,  13,  205 
Ditetragonal  prism,  90 

pyramid,  87 
Ditrigonal     prism    (hexagonal), 

134 
Dodecahedron,  rhombic,  55,  177 

pentagonal,  68 

tetrahedral-pentagonal,  78 
Dodecants,  25,  105 
Dolomite,  210 
Domes  (del),  36 

brachy,  148,  174 

clino,  163 

macro,  148,  174 

ortho,  163 

Enantiomorphous  forms,  66 
Epidote,  167,  198,  202 
Epsom  salts,  156 

False  planes,  212 
Fergusonite,  99 
Fixed  forms,  57 
Fluorspar,  192,  206 
Fundamental  circle,  232 

form   (=  ground-form),   47, 
83,  106 

Galena,  61,  211 
Garnet,  61 
Gold,  206 
Goniometer,  21 
Gypsum,  186,  210 
Gyroidal    hemihedrism  (isomet- 
ric), 63 

Habit  of  crystals,  11 

Hematite,  130,  186 

Hemihedrism  (def .),  39 
gyroidal  (isometric),  63 
pentagonal  (isometric),  66 
tetrahedral  (isometric),  71 
trapezohedral     (tetragonal), 

97 
pyramidal  (tetragonal),  98 


Hemihedrism,  sphenoidal  (tetra- 
gonal), 100 

trapezohedral     (hexagonal), 
118 

pyramidal  (hexagonal),  119 

rhoinbohedral    (hexagonal), 
122 

sphenoidal   (orthorhombic), 
155 

monoclinic,  167 
Hemihedrons  (def.),  40 

apparently    holohedral,    64, 

68,73 
Hemimorphism  (def.),  42 

tetragonal,  103 

hexagonal,  140 

orthorhombic,  156 

monoclinic,  168 
Hemi-orthodomes,  163 
Hemi- pyramids,  162 
Hexagonal  system,  45,  104 
Hexahedron  (=  cube),  56 
Hexoctahedron,  51 
Hextetrahedron,  71 
Holohedral,  40 
Holohedron,  40 
Hypersthene,  154 

Icositetrahedrou,  55 

pentagonal,  63 

Ideal  crystal  forms,  14  [69 

Inclined-face  hemihedrism,   42, 
Inclusions,  213 
Indices,  26 

of  a  zone,  219 
Individual  crystals,  16 
Integrity  of  zones,  221 
Intermediate  axes  and  planes  of 

symmetry,  82,  106 
lodosuccinlmide,  103 
Iron  vitriol,  165 
Isometric  system,  44,  46 
Isomorphous  growths,  181,  201 

Juxtaposition  twins,  185 

Kernal  crystals,  13 

Levy's  system  of  crystallographic 

notation,  32 

Limiting  elements  of  crystals,  18 
Limiting  forms,  57,  76,  91,  114, 

150,  164,  175 


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